Hello there !
I had also a lot of trouble with the Remeha Avanta Plus 28c, and tried a lot until i finally could find the solution. In my case, the error code E4 was showing the most, code E5 not so often, and E6 was very rare.
It was hell, ongoing for 1.5 years now, and i felt that i should share the information for other users plagued by this monster.
To make it short > let your plumber do a full service ! Aside from the "normal" things you can check (clogged siphon, turning the plug in the outlet), he should take care of the following things :
1.) Especially the cleaning of the heatings inner parts is very important. Make sure he orders a full service kit with all seals necessary. Our set had 3 seals : 1 big red "rubber" sealing, 1 smaller red rubber sealing, and 1 small graphite sealing for the burner/torch. Then a new ignition plug, a new temperature sensor (additional, no old sensor was replaced), a sort of "heat insulation" for the combustion chamber, and some replacement screws. Also you should get some very fine sandpaper.
2.) The small cover of the burner (with the Remeha logo), where the middle sized red sealing is fitted, has three "noses" on the inner side, which keep the burner in its place. These noses were dirty, and we cleaned them with fine sand paper. Obviously the electricity has to "flow" partially over these noses, and the dirtier they are, the less electricity can flow. Resulting in wrong values, the main board seems to check many of them.
3.) The burner itself has a sealing on the bottom, which was initially made out of some mesh/web-like material. Meanwhile Remeha uses graphite sealings on this spot, not mesh anymore. Replace it accordingly.
4.) In our case, the ignition transformer was also defective. We replaced it with a new one. I would suggest to replace it only if the problems remain.
5.) Afterwards, let your plumber check and adjust the emissions/exhaust values at "full load" and at "partial load" - there are specific modes you have to start to get them running. Check the manual for more information about this. In our case they were slightly out of the allowed range.
That was "all" i had to do to get the monster back at work. I think the most important was replacing the seals, replacing the transformer, and adjusting the exhaust values. ABout the costs : I am not sure how expensive the parts are in the UK, but here in Germany i found the transformer for ~50€ on Ebay, and the service kit with the seals etc. came for 85 € from the plumber.
Although this thread is older already, i am sure somebody out there will be happy to find a (possible) solution for this nerve-stretching issue.