NO by RAW, YES by RAI

A Recent Update to the FAQ clarifies the Constructed Property, but Mending is unchanged.

Since many of the answers here were posted, the Starfinder core rulebook received an errata/FAQ update that updated the constructed property for androids. The constructed property now reads as follows:

"For effects targeting creatures by type, androids count as both constructs and humanoids (whichever type allows an ability to affect them for abilities that affect only one type, and whichever is worse for abilities that affect both types)."

In essence, if an effect can target constructs but not humanoids, it can target them. If it can target both constructs and humanoids, but has differing effects between the two, the worse of the two effects is the result.

The problem here is now just the Mending spell. This spell is contradictory, as its target line specifies objects, while its text specifies both objects and constructs. Its intention is clearly that constructs should be able to benefit, but the current target restriction means that technically they cannot.

Targets one object of up to 1 bulk

(...)

This spell repairs damaged objects and constructs, restoring 1d4 Hit Points. If the object has the broken condition, this condition is removed if the object is restored to at least half its original Hit Points. All of the pieces of an object must be present for this spell to function. A construct can benefit from this spell only once per day. Magic items can be repaired by this spell, but magic items that are destroyed don’t have their magic abilities restored. This spell doesn’t reverse effects that warp or otherwise transmute items, but it can still repair damage dealt to such items.

This spell may recieve errata in the future.

No (Probably).

Cheek pouches can only hold 1 cubic foot. Since 1 foot is the minimum length/height a Tiny creature can be (Table 8-1, p. 256 Starfinder Core Rulebook), it's likely the drone is too big.

Then you have to consider:

1. Ysoki have two cheek pouches with a combined volume of 1 cubic foot. Since the volume is divided between these pouches, it's extremely unlikely a ysoki can use up their cheek pouch volume with a single large object. Cheek pouches are designed to hold many small objects, not one really large object.

2. The drone would need to fit in the ysoki's mouth. Even with cheek pouches, that's extremely unlikely.

You might be able to convince your GM that your drone is compact enough to fit, but then you'd have to convince them that such a compact little drone is sturdy enough to hold weapons, armor, etc.