Remove the lower section of the faceplate on your master socket and plug the HomeHub into the test socket that is now revealed. Test it on there, though be aware none of your extensions will work.
If you get the same issues on the test socket try another filter and retest. If you still get issues then try a different phone plugged into the filter.
If after all this you are still having the same issue then the chances are it's either the HomeHub or the line. Working out which could be a problem. Might be difficult to swap the HomeHub to eliminate that. In that case it might be time to beat up BT.
However if everything works perfectly on the test socket the fault is likely to be your extension wiring or something plugged into an extension. So first thing is to check the state of the wiring to look for any damage. Especially if any cable goes under carpet or round the edge of a door frame rather than through a hole to get to the other side of the wall.
If the cable is OK unplug all extension phones etc. that you have. Test again to confirm whether it is working OK. If so plug things back in one at a time and test each time. When the problem comes back whatever you last plugged in could be the problem. Try it with another filter just in case it's the filter not the phone/whatever. I assume you do have filters on all extensions (unless you have fitted a filtered faceplate on the master socket).
Another thing to do if you have extensions and while the faceplate is off on the master. Remove any wires on terminals 3 and 4 on the back of the faceplate. Leave just 2 and 5 connected, Go round all extensions and do the same. The wire(s) on 3 and 4 can act like aerials for interference. Many modern phones don't need any connection to these terminals.
BT will probably tell you to try at least the test socket if you call them to report a fault. And be aware that if they do send an Openreach guy out to investigate and it's your home wiring at fault they won't fix that and will probably charge you for the call out (which could be the thick end of £100).