Yes. Your travel time between service users’ homes is working time, for which you must be paid.
Your rate of pay is a matter for your contract terms but as a minimum, your employer must comply with national minimum wage laws. On average, your basic pay for your ‘pay reference period’, after adding your time spent travelling, must be at least the national minimum wage.
Your pay reference period is based on the pay arrangements where you work. It will be a day if you are paid daily, a week if you are paid weekly and a month if you are paid monthly.
The only time when travel time between shifts may not count for the purpose of calculating the national minimum wage is where the gap between your shifts is long enough for you – realistically – to be able to travel home.