Back wheel moving to left side
Thanks a lot for the comments, they did help a little at the time but did not solve the issue fully.
At the end I found there were a few issues with the rear of that bike.
The freewheel was wobbling a bit and had to be changed. The bearings of the hub had to be changed and the axle was bent so that was changed, too. Finally, the dropout was a bit bent, too on one side. After changing/correcting all of these the issue did disappear. But it was costlier than I had hoped  Unfortunately I cannot be sure which of these was the main issue.