I found this question on a JLPT N4 exercise, What I knew about the てくる is to do something and go back or to become, but I don't really think both might make sense.
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Understanding 〜てくる as 'start to verb' is a good starting point, but that isn't the only way to understand it, neither is it the best. In this particular case, we would best be able to understand its meaning with かかる. My understanding of かかってくる in this particular sentence is either
received. In light of the full translation, this might make more sense.
My translation is as follows:
When I got a phone call from my manager, I was preparing to go on a picnic with my children.
Or in a less wordy sense:
I was getting ready to go on a picnic with my kids when my manager called.
I would suggest that you read up on 〜てくる a little more. Here is a good resource.