You CAN claim Capital Allowances on a used car. But not the 100% Annual Investment Allowance that you could claim for a van, laptop, office furniture or similar. Not because it's second-hand, but because it's a car. (There are special rules for cars with VERY low CO2 emissions but they do not apply to your car).
When you say "late April last year", do you mean April 2019 or April 2020? If you mean April 2019 - early in 2019/20, then you can claim in 2019/20 but if you bought in April 2020 then you will have to wait until your 2020/21 tax return to make a claim.
You can claim 18% as an allowance - this is the "main rate" for any item outside the scope of the Annual Investment Allowance - but you will still have to adjust for private use, so £5,200 x 18% = £936 but restrict by 30% for private use so relievable amount = £936 x 70% = £656 in the first year.
The remaining amount (£5,200 less £936 = £4,264 NOT £5,200 - £656 = £4,544) is carried forward so that you can claim 18% of £4,264 in your next year's calculation (subject to any restriction for private use in that later year, etc., etc.)
You do not have to claim the full amount if you don't need to (if you limit your claim now, then the amount carried forwards to be claimed in later years is increased accordingly). It does not matter at what point you bought the car in your accounting year, so long as you are in a standard full accounting year (not commencement, changing your year-end or similar).
You can also claim for fuel, repairs and similar - again, with a restriction for private use.
But you CANNOT claim the alternative mileage allowance on top of a Capital Allowances claim, because the higher initial mileage allowance rate includes a theoretical amount to cover the capital cost of running a car for business purposes.