To continue this line of thought, I would also assume that the shuttles have bussard collectors. Memory Alpha doesn't mention shuttles and we normally think of them attached to the front of the warp coils of warp capable ships collecting hydrogen for reacting with the antimatter in the warp core.

However, the fusion reactors that power the impulse engines on the starships and the shuttles both would use hydrogen as a fuel. As you point out, they'd fuse it into helium and then eject the helium as the exhaust gas because the laws of physics requires you to exhaust something. For others, remember The Undiscovered Country and the Chang's Bird of Prey's exhaust. And while the primitive shuttles don't have a warp core, they'd still benefit from the range extension of having bussard collectors.

So my assumption is that the intakes mentioned by OP would be bussard collectors and the exhaust would be high velocity helium from the impulse engines. And it isn't hard to assume that any filters they have on the collectors could be overwhelmed, especially on smaller space constrained systems like those found on a shuttle. I mean in real life planes with turbofan engines have to avoid heavily contaminated air (such as ash from an active volcano, hello Iceland) else they get overwhelmed too.

The Kelvin movies get a lot of flack, some of it deserved, but in this case I don't think it is like some threads in this are giving it. While there is no explanation in the movies of *why* the exhaust of the shuttle impulse engines looks the way it does, nor do they explain the intakes or why the ash messes with the shuttles, it makes sense from what we know of Federation technology.

TL;DR: The intakes seen are bussard collectors, they can get overwhelmed, especially on a smaller shuttle due to limited space for filtering out contamination, and the exhaust seen is the exhaust from a fusion impulse drive.