A wizard casts the Mage Hand spell. They then pick up a short sword and a shield. They then want to cast Burning Hands against an opponent. Can they use the active Mage Hand to provide the somatic component for it?
Mage Hand is a spell, whose effect reminds of a hand but with severely limited capabilities compared to a real hand. Imagine it, as it is depicted in many illustrations, more like a semi-translucent shape of magical force.
The spell description lists the exact range of actions that can be undertaken through it. Some class specialization (like Trickster Rogue) have access to extra actions but nowhere it's listed that you can use it to provide a somatic component for further spellcasting. Because it's not an action.
Casting a spell with somatic components require gestures and possibly handling of a material component. This must be done with a free hand (gloves don't count) or while handling the material component in the hand. By the rules you can't delegate this part of spellcasting to a minion, a familiar, a friend, an invisible servitor or any spell effect.
RAW, no. Would I allow it, yes. What I would say though, is that you have to maintain concentration on it of course, but you have to succeed on a wisdom check of 10 + two times the spells level. If you have benefits to maintaining concentration from abilities such as the warcaster feat, you gain them on this check. Also, remember that it costs an action to summon the hand. As well as this, you would loose the ability to cast spells that require concentration, so I would say that its balanced.