The short answer
You can ask HMRC for more time, and doing so is common and generally reasonable, particularly where records are complex, held by a third party, or where you need to instruct an accountant to review the position properly. HMRC caseworkers deal with extension requests regularly and are usually willing to grant a short, sensible extension.
The key is to ask before the deadline passes, not after. A request made in advance, with a clear reason, is treated very differently from silence followed by a late explanation once the deadline has already gone.
There's no statutory right to an extension, so it remains at the caseworker's discretion, but a well-reasoned request with a specific new date is rarely refused outright.

