United haven't won the trophy now for nine years.
Carrick told talkSPORT: "It hurts. I've come close, I've played in the final, been in semi-finals and, for whatever reason, it hasn't fallen for us and we haven't been able to win it.
"It hurts because this club should be winning the FA Cup, especially having had such a successful period over the last six, seven, eight years. For a club like this not to win it over that period of time is criminal.
"It hurts the players because not many of us have won it. Hopefully, we have still got time to grab it, whether it be next season or after that.
"It does get mentioned [among the players] because we are desperate to win it. It is what I grew up watching. The FA Cup was the game of the season when I was younger and such a special day. We have come close, losing to Chelsea in the final, but we want to go better than that."