Yes but you will have to allow approx 100mm above the light inside the cupboard to allow clearance for heat. If you dont the box of Honey Nut Wheetos will be on fire.
Ok then, how about a setup like this (excuse lack of drawing skills), where the front panel has the switch mounted on it, hidden when the cupboard doors are closed, and the grey fill represents mesh on either side?
It might end up getting hot inside the whole cupboard. A 10 watt halogen under cupboard light is enough to make a mess of any chocolate or butter stored inside the cupboard if used for any length of time.
You could consider replacing the lamp with a low energy one, which will dissipate much less heat, or extend your cowl up and through the top of the cupboard, like a small chimney.
It is a bit more expensive but may be a bit more suited to your application, and will also save you losing valuable cupboard space, and you will enjoy reduced running costs, and a much longer lamp life
When you say the lights will be above the sink, what sort of distance are you talking? I would have thought it unusual to have wall mounted cabinets directly above the sink.
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