The Women's Super League champions were chasing a second trophy of the season as they faced Birmingham Ladies in a match beginning at 3pm.
Cushing told BT Sport he had been with his wife at hospital on Sunday morning, saying: "She's in the start of labour at the moment."
City posted a series of pictures of Cushing going through a range of emotions on their Twitter page, captioned: " When you find out your wife's gone into labour on @ContiUK Tyres Cup final day..."
A follow-up Tweet showed Cushing walking into the Manchester City Academy Stadium
The match was being played at the venue, which is adjacent to the Etihad Stadium
".@nickcush has arrived..
but his third child hasn't! What a day for the boss! #mcwfc" the club tweeted.
Cushing had told the Women's Soccer United website ahead of the weekend he would face a difficult decision on whether to be with his wife, or with his team, if her labour and the match coincided
He said: "My wife is due with our third, she is actually due on Monday so if she goes over then there will be an actual dilemma
"Do I go to the hospital? Or do I go to the game? It's a big game though isn't it
If I could get to the hospital in time, then I would go but this is a big game."
Source : PA
Source: PA