Voltage supply plus ideal current source connected to capacitor in series
However, what will happen if there is both an ideal current source
along with a voltage supply \$\color{red}{\boxed{\text{in series}}}\$ with the capacitor.

A voltage source that is \$\color{red}{\boxed{\text{in series}}}\$ with a current source is relegated to obscurity. You might just as well remove the voltage source and replace it with a short-circuit. The current source wins every time and defines the current flow.
On the other hand, a voltage source in parallel with a current source is different - the voltage source wins and defines the voltage; the current source is of no-consequence and can be replaced by an open-circuit i.e. removed.