Kyel Reid had given City a first-half lead but a Nathan Doyle own goal levelled the tie and Clayton headed home before Bradden Inman and Chuks Aneke netted late on to put the result beyond doubt.
"It was good for Max to score for his confidence. He's a good player but he is only 18 and it has been a bit of a struggle for him lately so I'm delighted for him to come on and make an impact," Davis said.
"I was delighted with the result but I wasn't over impressed with the performance although cup games are about winning. We managed to do that convincingly in the end but I have to admit I don't think we deserved to win by the margin we did.
"Our first-half performances haven't been up to standard off late and for some reason we don't get ourselves going until I have spoken to them at the break."
Bantams boss Phil Parkinson said: "I thought the lads did really well against a really good team with a totally changed line-up.
"We took them all the way and the scoreline that will be in the papers is in now way a reflection of the game.
"We nearly got back into it but we got caught on the counter-attack but that is the risk you take.
"Our two centre-backs did really well for two young lads on a difficult surface.
"The surface was as frozen as I have seen and if both managers were honest we would have said that game shouldn't have been played."