Solved
Recently I found that for periods WordPress would become very slow, often taking more than a minute to do anything. This seemed to begin for no apparent reason; fine one minute and dead the next. Then it would come right and be fine for a while. Because the site relies on the theme and plugins, I didn’t want to just turn everything off or change theme.
So I started by looking at Apache and MySQL, fiddling a few settings and watching what happened. But no improvement.
I then backed up WordPress and started turning off less important plugins. This took a while as each click took a few minutes to get a response. When I turned off the Bliss Hit Counter as if by magic at all came back to full speed. I had only installed the counter as a temporary thing to get a rough idea of page hits.
If it stays fixed I’ll try enabling the caching and see what that does.
Through Google I found a lot of questions and comment from people with similar problems. For them it could well be something else as the standard approach is to turn all plugins off when a problem occurs. It may be that the Bliss Counter conflicted with another plugin or the theme. I don’t know, but for now it seems OK.