Artificers have to hold a tool or an infused item in hand to cast spells…
Tools Required
You must have a spellcasting focus—specifically thieves' tools or some
kind of artisan's tool—in hand when you cast any spell with this
Spellcasting feature (meaning the spell has an 'M' component when you
cast it). You must be proficient with the tool to use it in this way.
See chapter 5, "Equipment," in the Player's Handbook for descriptions
of these tools. After you gain the Infuse Item feature at 2nd level,
you can also use any item bearing one of your infusions as a spellcasting focus.
So an infused Armor (ie. armor with enhanced defense) shouldn't be suited, or is it?
But some armors (ie. Half-Plate) come with gauntlets and the whole armor counts as one infused item… so technically you constantly have infused items in both hands even if you hold nothing in it, am I right? Or do I get something very wrong?
Best Answer
You can't use worn infused armor as a focus.
Your analysis of the "Tools Required" section is correct. You must have an infused item in your hand if you want to use it as your arcane focus. If you're wearing infused armor, you're not holding it in your hand, so you can't use it as your focus.
Even if the infused armor includes gauntlets, you won't have all of the infused item in your hand. You'll only be holding part of it, so a strict reading of the rules will conclude that it's not eligible to be a focus.
Your DM may make an exception for flavor.
One of the great things about being an artificer is figuring out how to re-flavor your spells, describing how exactly they manifest via your mastery of magical substances and devices. (See "The Magic of Artifice" sidebar on page 11 of Tasha's Cauldron of Everything for reference.)
Because of this class emphasis on creative re-skinning, your DM may be open to an especially fun idea about how to cast one or more spells through your gauntlet. Discuss it with them outside your play session. They may choose to allow it, or or the two of you may come up with an alternative that you'll both find acceptable.
An Armorer's worn arcane armor can be used as a focus.
The Artificer's Armorer subclass can use worn armor as a spellcasting focus, but it can't be just any infused armor; it must be the arcane armor granted by the subclass's 3rd-level feature. While wearing it:
Note that this armor can hold an infusion just like any other armor. Furthermore, at 9th level the Armorer gains the Armor Modifications feature which allows the arcane armor to hold multiple infusions. However, these infusions have no bearing on the arcane armor's ability to be used as a spellcasting focus when worn.
Related: Can an Armorer Artificer wearing Guardian-model Arcane Armor cast a somatic spell without holding tools or having the Warcaster feat?
The cleric has a similar, though rarely used, ability.
It's interesting to note that, if a cleric's holy symbol is emblazoned on their armor, they too can use their worn armor as a spellcasting focus.
This almost never comes up, though, because most clerics use a shield with their holy symbol on it as their focus instead.
Related: Do you need a free hand for your holy symbol if you're wearing it visibly?