I live in Zone 6/7 and leave my pump and waterfall going all year. See my winterizing page at
http://www.fishpondinfo.com/winter.htm for more on the pros and cons of keeping the pump running. In Zone 5, I normally suggest to turn off waterfalls but a small pump near the surface can be used to help with aeration. Aeration is vital in the winter as, if there is ice cover, there is less gas exchange. An air stone can sometimes keep a hole open as well as a de-icer with the benefit of adding oxygen. I have both aeration and a de-icer in my 153 gallon pond and the waterfall going in the 1800 gallon pond.
A tarp over the pond will help keep it warm but it is dangerous if it prevents gas exchange.
Aeration can be created from a pump that just outputs or goes to a fountain or waterfall. Aeration can also be created from an air pump which is simpler and very effective.
Without aeration, there is less gas exchange at the surface but, as long as there is an opening, there will be some gas exchange.
In the summer, plants under the water produce oxygen during the day but actually use it at night so a pond with tons of plants can have low oxygen overnight, enough sometimes to kill fish. Thus, water movement and aeration are a good idea every day of the year.
At this point, I would not put the cover back on the pond. A cover should only be used it if it elevated above the water's surface and has outlets for gas exchange. An aerator of some sort is really also needed when a cover is used. It's late enough in the winter that you shouldn't need the added protection of a cover now.
Unless the pond is over 3 feet deep, the cooling of the bottom layer is really not a concern. My winterizing page has more on that. You can set the pump about a foot below the surface and have it broil the water there for aeration and water movement. Elsewhere in this forum, I have discussed with others, including KK who lives in the coldest place on earth that has an outside koi pond, that it's better to have water movement to add oxygen and prevent stagnation to worry about the temperature stratification in the water of shallow ponds.
For a pond of about 300 gallons (assuming your pond is 2 feet deep, that's the volume), a pump running around 700 gph is good. I prefer to turn over ponds twice an hour, especially smaller ones. Experts often say to turn over every two hours but that's really not enough to keep ponds in good shape. I have a 700 gph pump in my 153 gallon pond and a 2600 gph pump and a 700 gph (summer only) pump in my 1800 gallon pond.
Most algae will die in complete darkness. Some light will get through most tarps. Read over my pond algae page at
http://www.fishpondinfo.com/plants/algae2.htmGood luck!