The most common UK pest that might cause this damage is the larvae of the Plum Fruit Moth. These little caterpillars are about 15mm long and can be red, white or pinkish in colour. They enter the fruit at the top (close to the stem) in July and August. The black crumbly substance is waste that the caterpillar has produced while eating the inside of the fruit. The adult moths lay their eggs in June and July. We sell traps that can be hung in the tree from May to August to monitor the numbers that are present. See this link: Fly Traps. If significant numbers are present then it may be necessary to spray with an appropriate insecticide.
Any fruit with the caterpillars inside should be collected and destroyed this month. Make sure that fallen fruits are collected too. If they are left then the caterpillars drop from the fruits to the ground and pupate over winter in the soil beneath the tree before the moths emerge to continue the cycle.
I would also suggest that the customer tries something that I recently read. Apparently if you lightly cultivate the soil with a hoe around the base of the tree in early March and again in April, this will disturb the pupae and hopefully they will be eaten by birds.