Can I file on behalf of my child?

Please explain what you mean by "on behalf of my child."  Do you mean you want to claim a child as your dependent?  Do you want to prepare a return for a child who had taxable income in 2019?

Re: Can I file a 2020 return on behalf of my grandchild? my mom?

Please explain what you mean by "on behalf of my grandchild? my mom?" Do you want to claim these people as dependents? Or do you want to file tax returns for them?