I don't understand the meaning of the word "cast".
Specifically, for example, in this phrase:
"Cast the approval"
I've never seen this phrase before.
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It's fairly common to use cast in this context, but perhaps not immediately obvious what it means.
When we use the word "cast" as a verb, it's hard to put ones finger on a cover-all definition, but lets start of with the google definition:
Cast [literal] to throw (something) forcefully in a specified direction.
You may have heard the phrase:
"Cast your vote"
I bring this up just because it's a much more common use of the word, which has the same general meaning as cast the approval, which I haven't heard used as a phrase before.
Taking the literal definition we can ascertain the metaphorical definition by just replacing the verb (throw).
I.e. cast your vote would be to vote [forcefully] in a specified direction.
I put forcefully in brackets as I don't think this is necessarily true in the metaphorical definition.
Now, to answer your original question I would assume (although it's hard to say without further context) that it means:
to approve [forcefully] in a specified direction
This is why it feels unnatural. Approving is already a "direction". However, common phrases also include cast your mind back, and cast doubt [upon something]. So it seems that the rigorous definition has been warped with time to just mean that it is doing the verb which the phrase describes. Therefore I would say:
To cast approval is just to approve.
Of course, this is mostly speculation on my part. As you can see, definitions seem to not be very rigorous.