Alvaro Negredo bagged a hat-trick, Edin Dzeko scored twice and Yaya Toure added another as a swaggering City eased towards a place in the final at Wembley.
Wednesday's result came just days after an FA Cup humbling by Nottingham Forest, albeit that beaten team an inexperienced one, and Allardyce acknowledged the threat to his job.
"You cannot keep losing football matches, that is for sure, as a manager," Allardyce said.
"You cannot think it will be allowed to continue. I don't want it to continue that is for sure.
"We know the reasons why but in the end even those reasons won't save you. We have to get those (injured) players back as quick as we can and we have got to not lose confidence in our own ability and get out of this hole we are in at the moment."
Allardyce had earlier struck a philosophical tone when reflecting on "outstanding" opponents.
"We are very disappointed obviously, but to look at Manchester City's record at home this season and what they have done to Tottenham and Manchester United, and Arsenal - we knew they had to have an off night and we had to have a really good night from our point of view," he told Sky Sports.
"The fact of the matter was that Manchester City's finishing was outstanding, never mind the build-up play. Outstanding in every sense of the word.
"We have a struggling defensive unit at the moment because of injuries and down on a bit of confidence. We have a left-back [George McCartney] playing centre-half that we got on a free transfer and it is not fair on him playing against Negredo and Dzeko today, but that is how it is.
"We got Roger Johnson who has just come and I thought he did a good job. He did as well as he could even though we got beat 6-0, but that is the problem, we haven't got a fully-fit squad. We are in a bit of hole at the moment and we have to dig ourselves out.
"Now it is not as if they hadn't done this to anyone else, if this was a team we came to and they beat us 6-0 and they hadn't done that to anyone else this season you would say this is not good enough, but we tried our best against a side that has got outstanding players because they play outstanding money.
"What do you expect when you have got lads who are free transfers against lads who have cost hundreds of millions and are paid hundreds of millions of pounds in wages?
"You can do better than we have done but I need a fully-fit squad and have a little more confidence than we have today, but when they do Manchester United, Tottenham and Arsenal then sometimes you have to take it on the chin.
"It is not the right time for us to get beat 6-0, we wanted to do better in the semi-final but face facts they are a miles better squad than us and have miles better players than we have got and certainly when we want to compete against a team like Manchester City we want our best team out on the field and we can't get that out at the moment, so we have to wait until we can and try and refocus for Cardiff in just a couple of days."