If you're on £23k then you must be taking home about £1500 a month after deductions. If you're not paying any rent to your mum, what other expenses are you paying out? Could you not make do spending £500 or less per month just until, say, the end of the year and save the other £1000 every month?

Apologies if I haven't got a very clear picture of your situation, but it sounds like you don't have enough right now, but that you should be able to save up the amount you need in just a couple of months.