Hi,
Practically speaking, that's correct, a P45 is what you give to your new employer to let them know what your final pay and tax was at your last job, so that your new employer can operate the correct tax code. However, if the tax year ends before you begin working for a new employer, there's no need to inform them of what your final pay and tax was with the old employer.
Meanwhile, the P60 you've received will confirm your full PAYE pay and tax for the year, so you won't need the P45 for that either.
Your old employer should still be able to help confirm your final pay ant tax with them, though.
Our Income Tax team can also confirm that by contacting them below.
Income Tax: general enquiries
Thank you.