I'll go through the options you proposed and the community's other suggestions.
Hovering in midair: This is a pretty good word for anything being somehow in midair and not falling down. Something hovering can have lateral and/or vertical motion (like the "hovercars" and "hoverboards") or stay in one place, perhaps the more natural interpretation. It might be hovering under its own efforts, like a hummingbird or flyboard; due to some upward force, like a balloon over a vent; or by some apparently unexplained mechanism, like a magic carpet...
Next, these two seem to me to correspond to the image I think you were describing in your comment: an object (rocket or no) flies up into the air, reaches its apogee, and appears to pause for a second before beginning to fall:
Hanging in midair: As if it were hanging by an invisible thread. Can be used of objects permanently hanging, or ones that simply appear to be in this position for a second or two.
Suspended in midair: When you consider that "to hang by a thread" means "to be suspended by a thread", you see the equivalence of this expression. I'd say that it's slightly less imaginative than "hanging" and seems a little more readily to imply literal suspension.
I skimmed FumbleFingers' answer too quickly at first and didn't even notice that the following two words are the same by which hover is defined in MW.
Stoppage of motion
Stopping in midair: Suggests perhaps a plane engine that ceases to work while the plane is in flight. The plane will not be "stopped" in the sense of motionless, though. In that other sense of "stop", there seems to be a longer timeframe implied, hence a physical impossibility.
Halting in midair: About the same analysis as "stop".
Holding in midair: "Hold" has two senses that could fit here: "halt" (but this is an archaic sense of "hold") and "keep (one's course)", like a plane. In fact, I'm probably thinking of the latter because of expressions like "holding pattern". However, there's no stoppage in midair for this one.
Freezing in midair: Much like the second sense of "stop", this implies a longer timeframe and hence a physical impossibility. It certainly doesn't imply "having difficulty keeping that position"; someone "frozen" is understood to be paralyzed and powerless.
Staying in midair: A little awkward, but comprehensible. It doesn't imply a lack of motion, however. It just means that the thing stays aloft and doesn't fall to earth.
(Speaking of which) Aloft: An adjective, but possibly a useful one for your purpose. It implies neither stopped nor continuous motion, just that something is airborne.
Continuity of motion
Coasting through the air: Moving laterally; whether slowly or not, this is not under its own power but on residual momentum. (When you stop pedalling on a bike and just keep rolling forward, you're also "coasting".)
Drifting through the air: Slowly moving laterally, possibly also vertically but mainly laterally (lest we should just say it was "falling").
Floating in/through the air: Floating in the air is about the same as drifting. Both connote the exact opposite of "having difficulty keeping that position". The sorts of things that float in the air are fireflies or "white cottony stuff", which should give you a sense of the effortlessness implied by the term.
I think those are your main options besides a paraphrase, or an ontological reexamination. For example, when you say in your comment that bees and helicopters "hover but don't have difficulty doing so" and so don't fit the criteria, I'm not sure what other example you have in mind that is actively expending a lot of energy to maintain its place in the air. A hummingbird, perhaps, but who knows if a hummingbird finds its job harder than a bee does?
If we were talking about futuristic technology, one expression we might adapt for such a purpose is "treading water"... I could imagine saying that someone is "treading air" to stay in one place, while wearing some special equipment or other.
To take another stab at what you're looking for, one very specific term for talking about having difficulty staying in the air is when a bird beats its wings rather than flaps them. A bird that flaps its wings is performing a lazier, more casual action, one performed with larger wings, perhaps — whereas a bird that beats its wings is expending more energy. For example, an injured hawk trying not to fall to the ground might "furiously beat its wings". (Incidentally, hummingbirds also beat, not flap, their wings.)
Meanwhile, two good all-purpose terms for doing something with difficulty are work hard and labour. If you're in some kind of pedal-powered flying machine, you might well be "labouring to stay in one place" in the sense of holding your position in the air. You wouldn't be motionless, of course, but you might nevertheless keep from locomoting.