My first thought when reading the title was to have a Planetary Debris Field - around our own planet, rocket launches are gradually getting riskier due to the amount of space junk in orbit around the planet. At the speeds that rockets travel, hitting even a small piece of something that broke off of a previous mission could be Very Bad for the current mission.
But in your question, you totally upped the ante, stating:
There was a planet, but it was destroyed by their crash-landing in the star system.
Bam. You have a literal planetary debris field right there. That is, the debris of an entire planet, all smashed up and still in orbit around the cloud's common centre of gravity. And since your ship's crash-landing into the planet is what kicked up all this dust, the mothership also happens to be right at the centre of this debris field.
Smaller scouting ships can navigate through the debris field in relative safety, mining the rocky bits or moving past the cloud to explore other planets in this solar system or whatever, but the mothership is stuck in the cloud for the foreseeable future. All spare power on the mothership is being pumped into keeping the shields up and hull intact, so there's:
- No room to accelerate (and/or warping space to make your technobabble-FTL-tunnel will also warp a continent-sized asteroid into your window),
- More pressing need for raw materials elsewhere,
- Not enough power to fire up the engines even if they were working
And obviously, FTL drives in this universe are necessarily big. You can't exactly kit out an existing one-, five- or twenty-person craft with a warp core, drive, and engines. Even the computer programs needed to turn all that on properly are way more advanced than what the smaller vessels can run. Couldn't that program run on the mothership to remote-control your puddle-jumper? Theoretically, maybe, sure - but that would involve opening a FTL corridor since FTL is necessarily faster-than-radio, and opening an FTL connection just takes us right back to the first problem. That's also why they can't send a distress signal back home to confirm their current location.
So even if they could turn on the FTL engines, doing so would be a suicidal move. Your big colony ship is stuck and not able to move anywhere or communicate with anyone outside this solar system anytime soon. Better get those engineers working overtime so they can complete their Von Neumann mining probes. Hopefully they can get 'em going in time to dig us out of this mess sometime within our colonists' lifetimes!
The colonists getting their hands on a new FTL engine would be an FTL engine (ideally, one accessorized with a compatible ship) that's not mired in middle of the disassembled planet. Such a find would allow them to escape this solar system years sooner than they ever could have without it.