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I think you shouldn't make suggestions that haven't been thoroughly calculated. The 47 uF capacitor would create a delay in a high seconds up to minutes range, hardly useful. The previous suggestion by vfone in more in the useful range, although I'm sure about the resistor value.i think you remove the 20k resistor from the base of the transistor & put a 47uf electrolytic capacitor
at the base of the transistor. replace 4.7k with 2.2k & place a 100uf electrolyte across relay. i believe that it
will solve your problem.