If it's the clear plastic strip between the light and water, I'd be worried about evaporation and condensation collecting on the bulbs and dripping on the sockets.
i would just go on ebay and buy a cheap hood to replace the broken one and call it a day and not mess with exposed bulbs exploding from water getting spashed on them. why risk it when to fix it properly is a very cheap option.
Isn't there another plastic piece on the actual strip light that came with the hood. I got one of those (or very similar by AGA) and it came with the plastic piece that broke on yours and another one. If yours did you should be fine with one layer protecting it. If not, I'd follow their directions because its better to pay the 20 bucks or whatever instead of losing your fish.
or just go a replace it with a glass top. they look nice, aren't priced too bad, and give you the option of adding more light later if you want more live plants.