CPL privileges do not state they can be exercised at night - but to gain a CPL night training is required.
Not always. I have a CPL, but I have never had a night qualification or rating due medical restrictions.
Not now possible, I believe, under EASA:
FCL.315 CPL — Training course
An applicant for a CPL shall have completed theoretical knowledge instruction and flight instruction at an ATO, in accordance
with Appendix 3 to this Part.
Appendix 3 requires 5 hours of night instruction, including 5 solo Take Offs
and Landings - If a medical does not allow PIC at night then the training
requirements for CPL issue cannot be met.