Victoria
There really is no definitive answer to this
Part of the answer depends on how closely linked your care and mobility needs, for some conditions there is a clear deliniation between the two, for many there is some level of crossover.
If you believe you can address this in a letter then it is worth doing, if not, then you may have to accept the possibility that your care needs will be re-assessed. Statistically (I don't have the figurs to hand), the DWP will accept your change of circumstances without further action, but there still remains a risk.
Good luck
Gordon