If the relay is a "DC" type and you have access to rewire the connections to the coil, the attached circuit may work for you. The on-time is a function of the relay coil resistance and the value of the capacitor. Approximately: C(in farads) x R (in ohms) = on-time (in seconds). The diode (D) a garden variety rectifier such as 1N4001. The resistor R is a non-critical value that discharges the capacitor after the switch opens...say, anything from 1X to 10X the coil resistance.
If you don't have access to the existing relay coil, you could add this circuit with a second relay.
Ken