 a few examples following.
edit 1
It dont seems to be very common, I was trying to found other usage by searching for the same term, I found out this document C Ring Project which mentions admissible function at page 50, but it takes the definition from Mendelson who is the author of the book Im following.
Some books seems to use "n-admissible" definition too, as examples we have
The Number Systems Of Analysis and Theory of Radicals

A: What you wrote is not quite correct. During the proof we do not yet have an $F$ that we could restrict, thus one cannot "specify a restriction of the function $F$". But if one inserts the word "hypothetical" into that, then one gets a correct (albeit not yet precise) description of the concept. The underlying idea is that if a function $F$ with the desired properties exists, then the restrictions of $F$ to initial segments of $P$ must satisfy certain conditions. The concept of $n$-admissibility makes that precise.
Conditions 1-4 say that the domain $A$ of an $n$-admissible function $f$ is an initial segment of $P$ containing $n$. Condition 5 is property a), and condition 6 says property b) holds as far as that makes sense, that is, on the whole domain of $f$.
Altogether, these are the conditions a restriction of $F$, if such an $F$ exists, to an initial segment of $P$ containing $n$ must necessarily satisfy.
Then the proof goes on to show that there is a unique $F \colon P \to W$ with properties a) and b), and thus $n$-admissibility is also sufficient to characterise the restrictions of $F$ to initial segments of $P$.
That proof is a (maybe the, discounting trivial variations) standard proof of the iteration theorem. That the concept of $n$-admissibility doesn't seem very common is because many authors do not give it a name. One can use the idea without naming it. Having a name for it adds convenience at some points of the proof, but at the (not very high) cost of introducing a new name.
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