延長できますか？って英語でなんて言うの？

"Can we have a bit more time,please?"
This is a polite way to request a time extension.

"One more hour ,please."
Here you are giving them the specific time extension you would like to be granted

A:Can we have a bit more time,please?
B: We are about to close.How much time do you need?
A: One more hour
B:One more hour then.After that we are shutting you down.

shutting you down= to force you to stop.

To carry on = to continue doing something.
"If you carry on making that noise I'm going to get very angry!"
To extend something = to make something longer.
"The shops in the city centre are extending their opening hours and will now close at 8pm."

There are some different ways you could ask to extend your time when you want to continue singing in a karaoke place or bar. If you would like to be more formal, the first two options might be better: "Can we extend our time?" and "Can we keep going a little longer?"
If you would like to be a little more informal, you can say something like: "We want to keep singing a little longer. Is that okay?"
It is always important to use the word please as well so we can make sure we are polite in whatever situation whether it is formal or informal.

Asking to extend your time, is asking for more time or asking if you can sing two songs instead of one, is also extending your session.

"May I" or "Could I" are very polite forms of asking a question.

"May I continue" is a polite was of asking a question. In this sense you would be asking to sing more songs than you originally planned on.
"Would it be a problem" is asked to confirm every thing is ok, and that no one would get upset by the action that you are taking.

"May" and "can" are slightly different as one asks for permission while "can" refers to the ability to do something. Native speakers tend to use these words interchangeably. Using these expressions explain that you want more time than the usual session. It is always good manners to say please when asking for something and it also conveys your sincerity.