City won 2-1 at Wembley thanks to Ilkay Gundogan's double as Pep Guardiola's team took another step towards a historic Treble.
Gundogan scored his first goal after just 12 seconds -- the quickest ever FA Cup final goal -- but his second came from a weak shot from the edge of the penalty area which De Gea failed to stop.
"In this moment, I don't want to talk about such issues of criticism because we played all a great season -- including David De Gea, he played a fantastic season," Ten Hag said.
De Gea is expected to stay at Old Trafford with negotiations about a new contract at an advanced stage but asked specifically about De Gea's kicking, Ten Hag accepted there must be improvement.
"Say it like this, we are in the right direction," Ten Hag added. "But there are occasions in the game, issues in the game we have to improve, definitely, if we want to make the next step and win trophies."
Following defeat in the FA Cup final, Ten Hag is facing a crucial summer as he looks to continue his rebuild of the squad.
Transfer plans have not been helped by uncertainty over the club's ownership, although a decision is expected sooner about whether British billionaire Sir Jim Ratcliffe or Qatari businessman Sheikh Jassim have been successful in their attempts to take over.
"I have only one plan, that is to improve this club and to improve this team," Ten Hag said. "I have my ideas and I already told the club what we have to do for that but I have to work with my staff, with my players to be better next season.
"It's not about getting players in and they do well. It's about we work all season and day by day to develop the team, progress the team. I'm very proud where we came from in the start of the season."