I can select it when clicking the track number, but the clip immediately clears off selection when left click to drag it. I then got into Track Properties by right clicking and tried to change the Start time, but the clip ends up at a totally different time from the one I type in.
 
Any ideas?
 
Thanks, Sylva.

well, you say, can't select and drag audio clip to a new position in track.
 
Why are you clicking the track number? That has nothing to do with clips.
 
Just click on the clip and drag it. You might be missing the correct snap settings.
 

And make sure you are using the correct "tool" in the upper left corner are the tools and you need to use the arrow for selecting. As Tim say's make sure to use the correct snap setting to have the track placed correctly in time. 

Could the clip be locked? Check clip properties to find out.