Verified answer

声がかかる means “receive an invitation” or “get invited”

声 means voice, かかる is large meaning of receive.

Altogether “receive a voice” means “receive an invitation” or “ get invited”.

Grammatically, we should say “声をかけられる” but that idiom has been taken by meaning of “receive an inducement” already.

So we use the idiom 声がかかる as “receive an invitation” or “ been invited”.

Your context, Chotto shiriai kankei de hitodebusoku no toki koe kakarun desu yo.

Means, “Well, I get invited(to work) by the people I know when they are short handed”.

The best answers I could find are the following:

1 〔かけ声がかかる〕

「いいぞ!」と声が掛かった

Somebody called “Bravo!”

2 〔特別のはからいを受ける〕

社長から声が掛かってそのプロジェクトチームに入った

He was added to the project team on the recommendation of the president.

From a few webpages, it seems to be an idiomatic expression for a recommendation from a superior.

Hope this helps some.

Edit: Thanks for the clarification Buddha