Yes. You are entitled to at least the National Minimum Wage for your time spent travelling between visits to the homes of different service users unless, perhaps, there is enough time to get home between each visit. It is up to your employer to organise your shifts in a sensible way to take account of the travel time.

You are entitled to at least the National Minimum Wage for this travel time regardless of what your employment contract terms say, because the right to be paid the National Minimum Wage is a statutory right.

Securing members their proper pay for their travel time between shifts is a key campaigning issue for many trade unions, so speak to your union to see what can be done.