There are several particulars that must be addressed in detail:
- Can the Mage Hand carry a creature? Yes, if it is under ten pounds.
- Does the polymorphed beast shape have the mental acuity necessary to control the Mage Hand? Depends on beast chosen, RAW is vague on this point and it is ultimately up to the DM. This is the most significant barrier to this maneuver.
- Flying. Is unfettered flying in this situation available with the Mage Hand? If the DM rules that the polymorphed shape can control the Mage Hand then according to RAW, the answer is yes.
Can the Mage Hand carry a creature?
There is nothing in the Mage Hand spell description the limits what you can carry other than (PHB 256):
The hand can’t attack, activate magic items, or carry more than 10 pounds.
This segment of the spell description:
You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.
In no way limits what you can carry. It is simply describing what you can do with objects. There is no exclusionary language.
It's worth looking at Mage Hand Legerdemain an Arcane Trickster Rogue's special feature gained at 3rd level, to see if there is any additional information there.
Mage Hand Legerdemain offers these extra features (PHB 98):
... when you cast mage hand, you can make the spectral hand invisible, and you can perform the following additional tasks with it:
- You can stow one object the hand is holding in a container worn or carried by another creature.
- You can retrieve an object in a container worn or carried by another creature.
- You can use thieves’ tools to pick locks and disarm traps at range.
This language explains what a rogue can do with the hand and in no way excludes any action by the standard Mage Hand cantrip. This is an explanation of precise and stealthy actions that can be done with the Rogue's Mage Hand Legerdemain while the hand is invisible. They are a huge upgrade from the simple tasks enumerated in the basic spell description. Again, there is nothing that excludes carrying.
Lastly, the 13th level Arcane Trickster Rogue gains these additional features with Mage Hand (PHB 98):
... you gain the ability to distract targets with your mage hand. As a bonus action on your turn, you can designate a creature within 5 feet of the spectral hand created by the spell. Doing so gives you advantage on attack rolls...
This last entry on Mage Hand also has no limitation on carrying. Once again it is describing what can be done, which in this case is precise control of the invisible hand during combat to target your adversary.
In all the entries on Mage Hand in the PHB the only restriction on carrying is the ten pound limit in the spell description.
Considering the examples presented in the spell description and that the Mage hand can't attack, I would suggest that the creature the hand picks up must be a willing creature under ten pounds, which works just fine for your situation.
Can the polymorphed beast shape control the Mage Hand?
The Polymorph spell indicates:
The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality.
The creature is limited in the actions it can perform by the nature of its new form, and it can’t speak, cast spells, or take any other action that requires hands or speech.
Considering you will have the intelligence of your new form, can a creature of 1-3 INT control the Mage Hand? Maybe 2 or 3 INT like a rat or cat. 1 INT, a frog? Most likely not. This would be the DM's call. I've seen different DMs handle the mental acuity aspect of Polymorph in different ways, but I'll leave that to another conversation. However I will include this quote from Jeremy Crawford from this related convo not as a definitive answer, but as food for thought:
A spell doesn't erase/suppress your memories unless the spell's text says it does.
When the text of a spell, like polymorph, says you retain your personality after a transformation, that's a terse way of saying, "You're still you, despite the radical changes you undergo."
Ultimately, if the DM says you're not smart enough to control the hand this maneuver won't work.
While completely unfettered flying may seem unreasonable with the Mage Hand there is nothing in the spell description that would limit this:
A spectral, floating hand appears at a point you choose within range. The hand lasts for the duration or until you dismiss it as an action. The hand vanishes if it is ever more than 30 feet away from you...
The only limitation for use is that the hand must be within range (30 feet) when cast, and it must remain within 30 feet of the spell caster for the duration. There is no limitation in the description as to how high off the ground the Mage Hand can be, as long as it is within 30 feet of the caster.
For example if a warlock casts the Fly spell on you and then you cast Mage Hand the hand will be able to fly with you. Likewise if you are at the edge of a chasm that is 1000 ft. deep and 20 feet wide, you can cast Mage Hand and the hand can go over the 1000 ft. chasm to pull the pin that releases the draw bridge across the chasm.
There is nothing in the Mage Hand spell description that limits the height at which the hand will function. If the mage hand is carrying the caster it is clearly within 30 feet and can go to any height and any distance at the rate of 30 feet per round. Although it may go against common sense, it is clearly within the limitations of RAW that you can fly unfettered as a polymorphed small creature being carried by the mage hand.
A final consideration might be is there a more efficient way to do this? Sure, but where's the fun in that? Also, being a non-flying creature like a rat, is far more useful for crawling into small chinks in a wall, but not so good at getting across the quipper infested moat. Being carried by the Mage Hand opens up a lot of possibilities.
Still, it's worth asking yourself,
Could I just polymorph into a bat and fly?