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.
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.