Hi Chris,
There is a capital gains calculator here:
Tax when you sell property
If the disposal value is less than the allowable costs results in a negative figure, then there is a loss.
Private Residence Relief is not required.
Where the opposite occurs and their is a gain, then Private Residence Relief is deducted from the gain, to a point where there a reduced gain or no more gain.
Private Residence Relief cannot reduce the gain to result in a loss.
It can only ever reduce the gain to £0.00.
Have a look at the guidance on private residence relief.
Thank you.