Because you are essentially an employee, the rules for claiming expenses against your employment income are very strict.
Where your employer does not reimburse your expenses, in general, any expenses which are wholly, exclusively and necessary to the performance of your role can be claimed via your self -assessment return or a P87 Form. This means that any person performing your role would to have to incur the expenses and that the expenses are incurred while performing your duties rather than putting you in a position where you are able to perform those duties.
However, if your end hirer reimburses you for expenses which are over and above your normal daily or hourly pay rate, these expenses may benefit from tax relief via the payroll if they are not part of your normal commute. In other words, if they relate to offsite or ad hoc visits to other work sites.
If you are a mobile worker you should also be able to benefit from tax relief via the payroll. It’s worth noting though that expenses incurred solely through your work can still be claimed on a tax return through HMRC, (if not reimbursed by your employer) for example, travel costs to clients, or if you have to pay for protective clothing or subscriptions necessary to your role.