Okay you have a number of options:
1. Try this link. Its for Win 7 but the principle is the same:
How to Show Day and Month Names in System Tray Date in Windows 7 - Tweaking with Vishal
2. If you right click on the task bar you can check if the task bar is locked or unlocked. If it is locked then unlock it. Go to top edge of task bar, left click hold down and drag up to make the bar two heights. Date should display permanently below time now.
3. Click on taskbar properties and select 'keep on top' if it is already selected then deselect and select it again. It's a well know XP bug. Happens commonly when your system is slow. Time to upgrade pc?
Some good advice here on how to speed up your system and fix the clock issue:
Go to System Properties and click on the Advanced tab then on the Performance button.
Uncheck:
Fade or slide menus into view (most important, as the time it takes is very annoying)
Fade or slide tooltips into view
Fade out menus after clicking
Slide open combo boxes
Slide taskbar buttons
Smooth scroll list boxes
You can also speed up your menus by unchecking "Show shadows under menus" too. Sure, it's eyecandy, but you really can't notice the difference.
More speed can be had if you uncheck "Show Shadows under mouse pointer"
Trust me, you wont notice that this stuff is gone, and your system will feel MUCH more snappy.