Mancini made the switch, with scores level, in the second half but promptly saw his side concede twice in 10 minutes and now sees his side rooted to the bottom of Group D.
City have switched to a 3-5-2 system on numerous occasions this season, but Richards revealed that the players are not comfortable playing in the formation, due to a lack of practice in training.
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"The players just want to play. It's a hard system because we're not used to it but I think the players prefer a 4-4-2 but he's the manager and we do what he says."
Richards praised Ajax's performance but admitted that City were expecting to win in the Dutch capital.
Defeat leaves City able to reach a maximum of 10 points, the same total that they finished on last season when they eliminated from the competition at the group stage.
Richards added: "I think we got beat by a very good Ajax side tonight. First half, when we went 1-0 up, they were probably the better team.
"But in the second half we just crumbled. Very disappointed with the lads' performance tonight.
"After last season when we got 10 points but didn't get through and when we got drawn this group, we knew it was going to be a tough one.
"But we were disappointing today, I thought we were going to come here and get the three points but it's not to be. We'll come in tomorrow and we'll see where the group's at.
"It's hard because, no disrespect to Ajax but you want to come here and win. We didn't do that tonight, but we'll look to the next game and see what we can do."