There are lots of components in the tape transport mechanism like what Bill mentioned can cause speed variation.
In play mode, use your finger and feel how much torque you get on the take up reel axle. If there is not enough torque, contact wheels in the tape transport mechanism can wear off from age or slip when it gets harder. The pinch roller rubber gets hard and cause all kid of speed variation. Normally I use a 1000 grit emery paper and sand a fine nano layer in play mode taking care not to sand the capstan or other parts. It is always better to replace those pinch rollers. When the pinch roller gets hard it can shrink also, which would give less pressure on tape against the capstan. Inadequate pressure means slippage and speed variation.
In some cases brushes on the motor can wear and cause speed variation. Also the tape speed adjustment trimmers too can cause problems. One of my decks had this speed variation, and when I turned the trimmer slightly forward and brought it back to the original position, rectified the problem.
Electrical contacts that activate play mode can also cause issues, so also loose or worn out belts or gears in the tape transport mechanism.
It will not be a simple fix if the contact wheels or gears are hidden inside the deck. It might take some time to get inside the deck assembly and replace them.
By the way what is the make and model of your boombox?