Oh dear. What a mess (and I'm not referring to the litter in the park!) "The council employs a handyman on a self employed basis..." as you're probably aware from the fact that you have raised this query, is a misleading statement. The council may either employ an individual (with full employer's responsibilities) or engage a contractor (on a single task or ongoing basis). You refer to "the need to agree to his increase in pay..." which would undoubtedly be a matter for the council (or a delegated committee) if the handyman is employed, but not necessarily for a self-employed contractor submitting invoices for work carried out. Incidentally, the submission of invoices does not define the relationship. There are many tests of the employed vs self-employed relationship, but in your case the key determinants would include whether you provide any tools or equipment and whether your handyman has the freedom to ask somebody else to cover for him if he is absent.
Check your standing orders and financial regulations for relevant issues and thresholds, as the increase from £75pcm to £95pcm takes the total contract value through the £1,000pa mark. Check delegated authority in your finance committee's tems of reference.
Ask the Clerk to put this on the next agenda to enable the council to clarify the legal status of the relationship, then propose a schedule of agreed works to be undertaken. You may also wish to re-advertise the role on a competitive basis to ensure best value is being achieved, although this may result in your current handyman walking away and no new applicants coming forward.
One further consideration - DBS check? You're employing an individual to work unsupervised around children.