A: 朝ご飯はなんだった?
B: こげん大きか茶碗に水のごたまずかおかゆと、梅のあのやおか...
Is ごた=こぐ(すごく) and まずかおかゆ=不味いお粥?
A: 朝{あさ}ご飯{はん}はなんだった?
B: こげん大{おお}きか茶碗{ちゃわん}に水のごたまずかおかゆと、梅{うめ}のあのやおか...
Not an expert on Fukuoka dialect, but I could somehow read B's line with no problems.
「~~ごた/ごたあ」 means 「~~のような」 = "(just) like ~~". You might be familiar with the somewhat similar-sounding word 「[如]{ごと}き」 that means the same in Standard Japanese.
「水のごたまずかおかゆ」=「水のようにまずいおかゆ」 = "rice gruel that tastes as bad as water" (It means the gruel is very thin.)
Fukuoka has ka-adjectives when the rest of the planet has i-adjectives.
Other dialectal words used:
「こげん」=「こんなに」
「大きか」=「大きい/大きな」
「やおか」=「やわらかい」
"What did you have for breakfast?"
"Terrible-tasting rice gruel as thin as water (served) in a bowl this big and (very soft pickled plum...)"