In answer to your specific question (Can I claim any of these back?) – the answer is no.
You now know why banks make so much money; you are not the first person to fall foul of exceeding your overdraft limit to discover what a costly mistake this is, and you won’t be the last.
Get the account back above the overdraft limit ASAP – the terms of the account will almost certainly charge for the time that the account is overdrawn, plus any additional fees applicable for payments from the overdrawn account. So even if you stop spending from the account, but leave it over the overdraft limit, it will still cost you dear.
See if you can’t find a better mortgage deal – when my last mortgage deal was ending, my lender advised me of this with plenty of time to shop around. In the end, I ended up swapping to another mortgage deal with the same lender (rather than going on their default mortgage – which would have resulted in higher charges).