The 2510 back light is notorious for running hot and melting the holder it is in, this can cause a problem as the bulb can get out of place and throw shadows.
Stains on the paper diffuser, it can be changed out but you have to tilt the front face of the radio out and remove the RIT, Squelch, RF gain and vol knob board to make room to take the display out. Not a hard job but a delicate one.
If the LCD display is dirty then the same removal is needed to clean it. From what you describe the display is not showing black spots in the display it self. If you could capture it and post a JPG here it would help.
I've taken a LED back light out of a display module and installed it in one of my 2510s, it has about twenty SMD LEDs and runs on five volts DC at 40 mA. I had to use a 5 volt regulator to drop the 12 volts down plus a resistor.
73 mechanic