Lawn scarified, thoughts on my next actions?

For context I’m in central England.

So 2 weeks ago I put down one of those 4 in 1 moss killer / lawn feed in we had a very wet winter and the grass was being overrun with moss. That’s now died and I’ve now scalped and scarified the lawn.

I’ve previously overseeded after this, but in England we often get storms in April and I’ve lost £50 of seed after it was all washed away so I’m a bit hesitant to overseed. I’ve also had a year where I didn’t overseed at all in the spring, and weeds grew in all the patches so I’m a bit more hesitant in deciding how to manage the grass this year.

I was thinking of getting some patch magic for the bigger patches in the photos, otherwise just leave it as is, let it grow and then just put down a weed and feed in May and maybe August to stop weeds from filling in any patches and then overseed in late September.

I’d appreciate any advice on my plans this year, thanks.