Can you evict a tenant due to rent checks being returned by bank for insufficient funds if they have a small child in the winter months?

You can certainly evict a tenant who doesn't pay the rent. You do not state whether the tenant pays the rent after the checks are returned. Usually, if the rent is late, landlords charge a late fee. Perhaps you can work something out with them regarding the rent. Find out why the checks are always returned: are they receiving income or assistance after the due date? Perhaps they are afraid to discuss it with you. It is sad to evict a family with small children at this time of year. But the decision is yours as to whether or not you want them as tenants. Also, you do not state whether this family receives Section 8 housing assistance. If so, then you need to abide by the rules of Section 8. There are specific grounds regarding eviction. Contact your local housing authority if this is the case.