If the vehicle stays at the test centre
GOV.UK says there is no fee for a partial retest when the vehicle is left at the test centre for repair and the retest is completed within 10 working days.
If you take the vehicle away
A free retest can apply when the vehicle returns by the end of the next working day and the failure involved only one or more items on the official qualifying list. If it returns within 10 working days in other circumstances, the test centre can charge a partial retest fee.
Can another garage do the retest?
Another garage can test the vehicle, but it will normally treat it as a new MOT and charge the full fee because it did not perform the original test.
Confirm before authorising repairs
Ask the test centre what deadline, fee and items apply to your result. Keep the refusal certificate and arrange the return date in writing where possible.
Official source
Check the current GOV.UK MOT retest rules.