I was thinking of the well known soft start circuit. You charge the capacitor via a low value current-limiting resistor, with the relay coil connected across the capacitor. At some point the relay coil will have sufficient voltage across it to be able to close it's contacts - this is the pull-in voltage (I think the proper term is the "must close" voltage) which is somewhat below the rated coil voltage. You want it to be high enough that the capacitor is mostly charged when it activates, but also low enough that it doesn't cut out at the low end of the supply voltage range
You arrange the relay contacts so that one set will short-circuit the resistor when the relay closes them. I was mistaken earlier, you don't need another set to keep the coil connected, So for the simplest version of the circuit you only need 1 set of normally open contacts. if you are also switching the motor out of circuit until the cap is charged, you need another set.
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The relay will stay energised as long as there is enough voltage across the cap to keep it so - this will be somewhat a lower value than the pull-in voltage.
The circuit that crutschow suggests will work equally well and provides a much better defined activation voltage. It will drop out at approximately the same point as it activates, but can easily be made to have a lower release voltage, like the bare relay does.
Given that you only have access to the + and - terminals of the controller, well, it's safe to assume that the capacitors you refer to are connected across the supply terminals. If that is true then you can still use the relay, just treat the controller input as if it's the capacitor itself. No need for diodes or anything fancy. If it's not arranged this way then you have two options that I can see. One is to use a timer, which really only needs to be a capacitor and resistor. It can be connected in exactly the same way, with the advantage that you only need a much lower voltage relay if you size the resistor appropriately. The caveat is that it's a timer, not something governed by local conditions. The other alternative is to use a current sensing circuit, but this would require more components and probably is rather excessive for what you want.