Thank you for the tip. Your suggestion was spot on. When I looked at the plate all appeared normal but, when I examined the plate using a magnifier, I saw a small crack. Apparently when I tighten the nut it caused the crack to open and I really couldn't get it tight enough. I made a new plate today and it seems to have cured the problem. Thanks again to all that offered suggestions.Also check that the plate that the tailstock is gripped to the bed by does not have worm grooves where it has been rubbed by the underside of the ways.
I had such a problem with the thread counter friction pad /plate being well worn in this manner .
I had to make a new one as I couldn't use the old one turned upside down due to the position of the threaded securing hole being too far to one side .