Yes, you can claim your child as a dependent on your 2023 tax return. You only need to indicate that the child lived with you for the entire year if you use a tax program such as TurboTax.
You will be eligible for the Child Tax Credit of $2,000 on the 2023 return. The only other deductions you can claim for the child will be for any medical expenses you paid for the child. Those are entered as itemized expenses on Schedule A. However, the total of all your itemized deduction must be greater than the Standard Deduction for your filling status to have any tax benefit.
For your partner, you will not be able to claim them as a dependent since they did not live in your home for the enter tax year of 2023. Assuming that you are not married.
