The people can hear me with the hand free kit and then suddenly can not hear me any more

Might the trouble be caused by the headphone jack?

Although I have not experienced any problems with my (different) headphones, maybe the handsfree-kit coming loose is causing your troubles?
Someone has cut out the cover a bit to allow for the jack to be pluged in more secure.
If your trouble is of that kind, that’s the only solution I know of, besides finding a handsfree-set, that fits.