The original sentence is: "Baby hares stay snug under a bush waiting for their mum."
This sentence is mostly correct but can be improved slightly for clarity. Here's a corrected version:
"Baby hares stay snug under a bush, waiting for their mum."
The only change is adding a comma after "bush" to separate the two ideas—where they stay snug and what they are waiting for.
Alternatively, you can say: "Baby hares stay snug under a bush while waiting for their mum."