I know you're not a native English speaker, but in many circles of general nerd culture in America and on the Internet, there is a phrase for what you're doing right now: your spaghetti is falling out of your pockets.
In all seriousness, though, you seem very anxious about your situation and you should take a step back and relax. You say you're good at improv, but it looks like you're already sweating bullets because your party got sidetracked instead of getting to the quest you wanted to take them on.
I'm going to put my direct answer to your question first and some general advice afterwards.
Your players probably won't miss their characters. Why? Well, you generated them, and even wrote up their backstories. These characters were never made by the players playing them, and while they might have grown attached over a few moments, I find it extremely unlikely that they'll shed tears over a party that lasted for four sessions... that they didn't even make themselves. Even if the characters are extremely cool and interesting, that element of personal attachment just isn't there, which allows the players to truly say that they helped create an exciting story with interesting characters. This obviously varies from person to person, but again, I find it very unlikely that your players would have grown very attached in this case.
Go out with a bang. You know that amazing epic encounter you were saving for the climax of the plot arc? Yeah, run it now. You'll have to make some tweaks because you obviously aren't quite there yet, but get them there as fast as possible. Feel free to kill off PCs or even have a TPK at this point; memorable deaths are often much better than "and then they lived happily ever after."
Here's a few things to keep in mind when starting your next campaign:
Your content will come to light eventually, and it will be good. The quests you've designed will always find a way to come forward. Even if your plot arc is entirely ruined by something the players did, you will be able to recycle the content you made but never ended up playing, and I encourage you to do so for your new campaign. The only things that are truly lost are "hard" materials, like NPC stat sheets, etc.
You're in control, and therefore, you set the tone. Sometimes, it is best to take a page out of Gygax's book; after all, this is your campaign, and you put a lot of effort into it. Obviously you shouldn't take the entire preface from the AD&D DM's Guide to heart, but there is a point where a DM should draw a line in order for there to be some kind of structure, assuming you want your campaign to go anywhere. If your players are goofing around and killing NPCs for no reason, or making light of important people in-character, then they should be ICly punished for it; reprimanded for insulting a nobleman, pursued for attacking innocents, etc. It is also very possible to play a serious game in character and laugh until you're blue in the face out of character. This frequently occurs in the Dark Heresy games that I've played and ran.
You had better get used to murdering your darlings. This is a phrase commonly used amongst writers and creative designers everywhere in the U.S. The phrase means that you'll have to scrap ideas frequently, including ones that you really, really liked, so you had better get used to it. The saying is intended for use in the writing, film, video game, and other industries where a publisher or producer oversees your work, constantly telling you what can and can't make it to the final product based on time and expenses. However, it works just as well for when your ideas can't make it to the game because your players did something insane. And, on a related note...
Plan less. I don't know how much effort you're putting into writing everything ahead of time now, but you might want to ease up on that. From what you're telling me about your role-playing experience, it seems like you've been playing in a "safe" and slow environment where you rarely, if ever, have to scrap or re-do material. This happens literally all the time in regular tabletop RPGs, thanks to the insanely unpredictable nature of 4-5 different people working together. It will save you a lot of anguish if you lay out a basic outline of what's going to happen and then add the details once you're sure the players will be arriving there next session, or maybe two sessions later.
It seems like you've learned a lot already OP, which is great, but scrapping a campaign after 4 sessions (and while your players are all enjoying it) is something you should really avoid. If everyone else is having fun, consider either shaking things up a bit and changing your own notes, or coming up with a way to set them back on track, which doesn't always need to feel contrived or railroad-y.
EDIT: Well, now that SevenSidedDie has made that edit to your post, there are a couple of details that I didn't quite catch before, no offense. Since you said you have a month between each session, it seems like you might be over-planning because you have a lot of time between sessions. Heck, you might even consider having more frequent sessions, if you can't stop yourself from overthinking it in the intervening months. If in-person is not an option, use Skype and/or Roll20.
It is a problem that Lost Mines of Phandelver is designed for 3-5 player characters (PCs), plus a DM. I'm playing Phandelver at the moment with a group. One session only two players turned up, and I discovered how deadly this could be with no modification. With one PC and no modification, you would likely not last past the first encounter. As far as I can see it, you have options:
1) Tone down Lost Mines of Phandelver
By 'tone down' I mean reduce the level of the encounters. This will be easier near the beginning (where you could just reduce the number of goblins), and more difficult as the game progresses.
I actually did this with the session in question - they were supposed to encounter 6 hobgoblins (rolling on the random encounter table at that point in the game), but I cut it down to two.
With bigger monsters it is more difficult, though you could 'wound' them, reducing their HP (hit points) and possibly some of their special abilites, and XP (experience points) commesurately, but that can get complicated and possibly lack verisimilitude, so there is always option 2).
I didn't do this to one monster (a grick) and both characters ended the session unconscious. Oops.
2) Play Phandelver, but have your girlfriend control two or more characters, and/or bolster the party with an NPC or two
In this solution, you can still play Lost Mines of Phandelver, but you could split the party between you and your girlfriend, with your girlfriend controlling two PCs and you controlling another two (or three if you want to use all five) as NPCs (non player characters - controlled by the DM).
This obviously has the disadvantage of complexity, but it can be fun nevertheless, and gives you and your girlfriend an idea of the different types of characters avaiable to play.
You could also combine 1 & 2 in the following way:
3) Start Phandelver with two (N)PCs, tone things down at the beginning, then add (N)PCs and enemies as you gain confidence
For example, your girlfriend could control one PC, and you one supporting NPC (just choose two from the pregenerated characters).
Spoiler, in case your girlfriend's reading :-)
Cut all encounters in half - so two goblins on the road, half the numbers of goblins in the cave, and either weaken the bugbear in Cragmaw Hideout and/or take his wolf away. Don't forget to halve the XP. Then at the end of Cragmaw Hideout, take Sildar Hallwinter with you and have him stay as a friendly (and tough) NPC. This presumes that your girlfriend will choose to go to Cragmaw Hideout first, but that is the most likely course of action, and you can always guide her back there from Phandalin if she goes there first.
Then as your girlfriend gains confidence playing, she could start to control another PC, and so you will have a tough enough party to face the later more challenging stuff without cutting things down.
For options 1-3 I strongly suggest you have a cleric in your party - parties with clerics last longer!
I haven't tried this myself, but for options 1-3 you might also want to read Play it Solo:Dungeons & Dragons Starter Set. Although it is geared to playing completely solo, looks like it might have some useful advice for converting Phandelver for playing with fewer players.
Or, for something completely different:
4) Find an adventure for one or two players from an earlier edition of D&D and convert it.
This is not my idea, but is described in more detail here:
Solo or 2 player adventures for 5e
You might have trouble doing this conversion just with the basic rules though - you would probably need at least the 5e Monster Manual, though I suppose you could just take the idea (and maps) from such an adventure and swap in monsters from the Starter Set.
Best Answer
Yes, just as in many good games (tabletop or otherwise), you are given a finite resource which you must learn to economize. It's all part of the game!
At low levels, it's pretty easy to run out of spell slots, you only have 2 of them so it is up to you to manage them and make the most out of what you have. Even as other spellcasting classes such as Bard and especially Warlock, you really need to think about what sort of situations you want to spend your precious 2 1st-level spell slots on.
Without spell slots, you have other options for offense but going into melee with your armorless self is less than advisable so your best bet is to use your cantrips just as you are doing right now. They might look a bit underpowered to you at first, but that is because these level 0 spells give up damage for a slight mechanical benefit. Shocking Grasp prevents your target from taking reactions(thereby allowing you to get away safely), Ray of Frost slows down your target by 10 ft, Fire Bolt sets things ablaze, even Sacred Flame negates cover. These cantrips also increase in damage as you reach higher levels (usually at 5th-level so it is still a ways off)
Don't worry, 2nd-level will come faster than you might expect. Then you'll have 3 1st-level spell slots to contemplate spending.
As a final note, and this may be nothing but if you find yourselves struggling for resources every session, remember: a long rest isn't a "session ender", you can take long rests in the middle. In my group, we have fun over the logistics like bedrolls and sharing them because one guy didn't have one.
There are also guidelines in the DMG on the experience earned in an adventuring day, found in page 84. I felt like I should mention this because it's odd that you've had 5/6 encounters and still haven't gotten to 2nd-level.