The end result is that the Mimic has no functional effect on a Mage Hand other than potential discovery of the mimic
A mage hand is not an object or a creature - it is a Spectral floating hand effect that can do a limited variety of actions. This doesn't mean it's not a thing, it's just not a 'thing' as defined by D&D 5e (as an object or a creature.)
Like Spiritual Weapon, Mage Hand isn't a physical creature/object and can't be used for advantage, etc. It can not attack, has no AC, no HP or any other statistics (it can not be attacked, either - unlike Bigby's Hand)
Utilizing the same intent, along with support like no statistics for the Mage Hand, leads to the conclusion that Mage Hand isn't something that can be attacked, grappled, or interacted WITH and would not be susceptible to a Mimic's Adhesive quality.
In order to be grappled, the mage hand would need to have statistics to make the Ability check - and Mage Hand does not have any.
If you allow for the Mage hand to be a "thing", then the Mimic would adhere to it, but it would not be grappled by it (since you can't grapple a mage hand.) The hand could still move, but would have a mimic attached to it.
The standard weight requirements for Mage Hand are still there, although this then becomes entangled in the issue of not having a stat block to break the "grapple" to become unstuck. If it can't break it because there is no ability score to break it, it's hard to allow it become grappled/stuck in the first place.
It just doesn't matter
However, even being stuck isn't an issue. There's no need to come up with an ability check or system for adhesion vs grapple to break free. If a Mage Hand is 'stuck', the best and most efficient way to free it is to cast a new one. Same cost (action), but with a guarantee to be free.
If you don't allow it to be a thing to get "stuck", then it would interact with the Mimic for no effect.
Ultimately, this question seems to be about would a Mage Hand interacting with a Mimic tip the players off about it's existence. If I was a player, I'd expect the Mage Hand to tip me off that the object wasn't an object. I could see a case (and provided one) for the other, but as a player it would be frustrating to be mislead in this manner.
BONUS ANSWER: Adhesive Effect
When in Object form, the Adhesive is secreted and is seems to be something that's "always on." The mimic has no control over it unless it is moving or changed form (MM, 220)
A mimic excretes an adhesive that helps it seize prey and weapons that touch it. The adhesive is absorbed when the mimic assumes its amorphous form and on parts the mimic uses to move itself.