In Dark Souls 2, what happens if you kill or don’t kill the Pursuer during the first encounter

Leveling up the Attunement stat will give you extra spell uses. There is also a ring that grants additional spell uses, it's called Northen Ritual Band

I hope this helps

You should be more concerned with killing him efficiently, instead. Here are a few tips for fighting the Persuer, and enemies like him:

First off, he uses a shield. While a shield offers a high potential for defense, it can be gravely detrimental if you understand the weakness it brings; it offers a blindspot in most, if not all, of his attacks. Wait for him to lunge, strike, or combo, then quickly roll towards his shield's side, and then punish him.

Secondly, the Persuer does not have any ranged attacks (Aside from a unique ability unlocked only if he manages to impale you while his sword is glowing blue, signified by a glowing blue tint in his eyes), but he does have a great gap-closer, which is telegraphed by him beginning to float, and then lunging toward you. Because it is easy to read, you can dodge it easily, and then having a chance to effectively punish him again.

Third, the arena is small, and only made smaller by the debris scattered around. If you need to grant yourself some more wriggle room, either break the obstacles, or bait Persuer into attacking them for you, and always be careful not to fall out of the arena.

Finally, always be sure that your weight load (Bottom right of equipment screen) is ALWAYS less than 70%. If you go over 70%, your characters dodge roll will be... more of a backwards belly-flop. Without your dodge roll, you cannot easily avoid enemy attacks. You are safer dodging an attack with light armor, than taking an attack head-on with the heaviest, beefiest gear in the game.

Boss fights in Dark Souls are about skills in dodging, reading enemy movements, and knowing when to attack, heal, or bait an attack. Some bosses have unique strategies, but many bosses can be killed easily with patience and well timed dodge rolls.