Hi,
there is absolutely no reason why a singer cannot present a performance with headphones. It may take some experience. This is only gained by practice! You can work with a handheld dynamic mic in the control room with the monitors low, and even try phase cancellation. You need to record WITHOUT compression to achieve this, or it won't cancel. You definitely make a rod for your own back if you go down this route. This game is all about control, and you lose a great deal of control this way. Do so at your peril. And you'd better be sure that the backing track is NOT going to change! You also cannot really alter the timing of any part of the vocal. Pandering to the whims and vagaries of an inexperienced singer is tantamount to folly. Give 'em an inch . . . . . Making a fuss about headphones is often an expression of "Red-Lights, Brown-Pants Syndrome". Concede too much, and you will create a monster, believe me! There is a very good reason why we do it the way we do! I prefer to only let really experienced singers do it in front of the monitors. - It is ok if you are Freddie Mercury or Bono!
I tend to try to keep the volume down in the headphones, and try to keep the vocal as quiet as possible, the singer listening to themselves in the room, with one ear off. When i am singing, I do not like to hear myself in the mix at all. It also helps to keep the compression as gentle as possible. I always record with some compression going to tape(well, Tools!). This makes the singer put some expression in, as they are working against the compressor. If you over do it, they stop working it, since it all sounds the same to them anyway.
Try a different pair of headphones. I find some types are very uninspiring, and some even seem to cause great singers to have pitch issues. As a good alround workhorse, that are bombproof, minimal spill, and sound pretty good, I recommend the Beyer DT150's. Closed back phones are essential in my book.
ALso, try sitting with the singer in the same room, and you monitor with one ear off too, checking them "in the room". This is by far the best way to judge a take in my book. It is so much easier to vibe them up when they are with you, sharing the whole experience.
Kindest regards,
ANdyP