1. It won't be covered by the 10 year NHBC guarantee, unless the property is less than two years old, when it might be covered.
2. You need a plumber with what is know as a G3 qualification.
3. What you have in your picture is not a boiler, it is an unvented hot water cylinder (hence the need for a G3 qualification). It is heated indirectly by the boiler, and will probably also have 1 or 2 electric immersion heaters for use should the boiler fail. However, I can't see a suitable electric lead in your picture.
4. I can't tell for sure, but it looks as if the silver coloured motorised valve, which is clicking, controls the flow of hot water from the boiler into the coil inside the cylinder. TH heat from this coil is what heats your hot water.
5. If you turn the requirement for hot water off at your programmer, the valve will probably stop clicking.
6. If the boiler is no longer heating the cylinder, see if there is an electric immersion heater fitted. Then, if you do want the water heated, find the switch for the immersion heater and turn it on.