Every time we flush the loo there is a loud vibrating noise which lessens when turning on cold tap.
How do we fix this please?

First, under the toilet is a valve that supplies toilet with water. Turn that valve OFF and then ON. See if the noise stops. If not, lift the lid and put your had on top of the FILL VALVE during fill up. See if that valve is vibrating. If yes, there is dirt inside. You would have to turn off water, take that valve apart, flush it under running water - then reinstalling it

Let me know how you did

Milo

Hello Hobby Hobbit,

Please follow Milo's advises and please tell us which fill will do you have see the images below, Thanks.

John