If he's self-employed (with company simply working as an agency that puts work his way), then the company can insist that he undertakes training at his own expense. (It's akin to me saying to someone who wants work tidying my overgrown garden that I won't employ him unless he can prove to me that he's a qualified tree surgeon. He can't expect me to pay for him to undertake such a course. If he wants the work he'll have to pay for his qualification).
If he's an employee, then the employer should pay the costs of the actual training but, unless the time spent on the course would take his average pay to lower than the National Living Wage (or his contract states otherwise), the employer isn't obliged to pay for the actual time spent on the course.
For further advice, your partner should phone the Acas helpline:
https://www.acas.org.uk/contact
(Several people on this site have reported that it can take a while to get through but that it's well worth the wait, as the advice is always accurate and helpful).