OK. After trying to figure it out and reading other posts, I did come across this suggestion on another forum - replace the BIOS battery.
http://www.pctechbytes.net/forums/showthread.php?t=15018
So, I did, but it didn't fix my problem. I then decided to find a suitable powersupply and replace it - I bought an Antec 500 watt powersupply as I read about it on another forum (several people really liked it). It finally arrived yesterday and today I switched the powersupply and have turned on and off my pc about 8 or 9 times, and everything seems to be alright.
So, I think, this can be a solution for some people, at least; as was the replacement of the bIOS battery for some. Good Luck