It sounds like you live and work in NY, but worked the first week in CT. Since you physically earned income in CT, you are subject to CT tax and have to file a non resident CT return.
The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state. You will have to file a non resident CT state return and pay CT tax on the income earned there.. You will also file a NY full year resident return and calculate tax on ALL your income. NY will give you a credit, or partial credit, for the tax you pay CT. So, there will be little or no double taxation, but you have the cost and hassle of filing two state returns. Do the nonresident state return first.
If your situation is different, explain.
