Most small hermetic single phase refrigerator compressors use a start winding that creates the necessary phase shift by its high resistance relative to its inductance, without a capacitor, and due to the thin wire used for the start winding it will burn out if continuously powered, so will typically have a high current PTC thermistor in series with the winding to provide a crude time delay mechanism to cut (most of) the power to it after start. TThis prevents rapid re-starting as the PTC thermistor must cool first, but it stays hot as long as the motor is running. There's also normally a Klixon or similar over-temperature trip in series with the common to both windings, to provide some protection if for any reason the motor fails to start. Both the PTC and the Klixon are usually integrated in the connector that plugs onto the motor pins.
If it didn't have a start or run cap don't add one!
If you need rapid restart capability, the PTC needs to be able to cool down while the motor is running. Adding a time delay relay to cut start winding power is one approach, either in series with the PTC or replacing it.