Add a relay to an existing switch
Your proposed solution works for half the problems.  The additional relay (that's what I recommend given the scant information about the load) is logically ORed with the switch.  If either is on, then the overall switch is on.  For the overall switch to be off, both the original switch and the relay need to be off.

Wiring the relay in series with the switch flips this around.  Now the on-states of the two are ANDed instead of ORed.  Both have to be on for the overall switch to be on.

If you are OK with one of the two above, then yes, just wire the contacts of a relay as you show (logical OR), or in series (logical AND) with the existing switch.