I'm not a sea angler but grew up in the area and can tell you Harrison Drive would be a good choice as fishing over the sea wall was always popular with flatfish, bass, eels and the the odd mackerel in summer.
Large areas go drastically underfished, especially between Harrison Drive and Moreton. The section of beach opposite Leasowe Golf Club looks absolutely mint and I would think would be good for a few fish.
There is, I believe, some decent fishing in the Dee Estuary around Hoylake and West Kirby too - both easily accessible via the local train service.
Hope it helps