I do not know where the OP found the phrase, but to me 催促(を)掛ける sounds very odd. It could be something that a native speaker wrote but these days there are many incorrect words and phrases used by the natives on internet...
I personally never use 掛ける for 督促 or 催促, they are something I use with する, and I've never heard/read 催促掛ける before.
(Sorry I'm being a bit off-topic-ish, as this is not what OP is actually asking about...)