I’d recommend either a Remington 870 or a Mossberg 500 in 12 gauge. The guns themselves aren’t expensive - about $300-$400 depending on the model. The utility of a 12 gauge is so much greater than that of a 20 gauge that I would recommend a 12 gauge with a recoil-reducing stock.
I do a fair amount of trap shooting myself, and I can tell you the Blackhawk stock makes a world of difference. Using a recoil suppression system like that makes 8-10 rounds (200-250 shots) no trouble at all, and the recoil is much less than even a standard 20 gauge. If you don’t mind the look of the stock, it’s a great option and makes shooting a 12 gauge quite comfortable.
I thought that pump-actions weren’t ideal for clays. Is that not true?