Of course, but it's important to know what exactly is causing the problem in order to find a solution. I recommend doing several recordings of the same piece or even segment of a piece. In the first take, do it with a monitor and no headphones. Listen to the recording to make sure everything is in tune. Then the next take, use a monitor still, but then have headphones over one ear which nothing coming through it, just using it to cover up an ear. As you are singing, alternate having one ear covered and not so that you will have a few seconds with it on and few seconds with it off while you are singing a phrase. Listen to the quality of the recording, hopefully your pitch will still be in good shape. You can then try this with having the sound coming through both headphones and monitor, but still alternating with having an ear covered and not while you sing. Check the pitch again. You can also try with the monitor very low, but headphones at normal volume and keep the phones on one ear the whole time, and repeat until there's nothing coming through the monitor.
If at any time your pitch suffers, then revert back to the last step and make sure it's secure before proceeding. This may be more helpful over a period of days rather than one sitting, so if you get to a step where your pitch does suffer, make note of where you stopped and then return to the previous step the next day and work forward. You're just teaching your ears how to listen differently and not get confused by the headphones. When I record I only cover one ear as well, I think that is probably the best way to do it.