Is it possible to officially backdate my employment start date with HMRC? I have been working part time for my current employer since April 2018 and I have just found out from my employer that he hasn't told HMRC about me (when I asked him for payslips). Can we declare with HMRC that I have been working since April 2018? I get paid in cash every month. I would also like payslips for every month. My employer and I are in the UK.
You need some way of proving when you started.
Usually on the beginning of employment, you would be given a contract and either this or a covering letter shows your start date with the company. Failing that, maybe an email?
With your employer not telling HMRC about you is a big red flag. I'm guessing they've been taking money from you each pay day "for taxes" but I'd also guess they not paying these taxes (meaning you could still be liable at the end of the tax year because HMRC could consider you self-employed if you don't have an official employer or contract) and your national insurance which is what contributes to your pension when you reach retirement age.
If you can prove when you started, you can contact HMRC yourself and give them the details although this could cause problems for your employer if you're not the only one who they haven't told HMRC about. Start by being firm with your employer about them telling HMRC about your from the date you started and getting payslips.