I have found in a reading the following theorems and definitions:
This is what I have understood about I framed in red: Using the recursion theorem I can define $f(n+1)$ as a single function $g(f(n))$, on the other hand using the strong recursion theorem I can define $f(n+1)$ as a single function of $n$, i.e. $f(n+1)=g^{'}(n)$. So I think that I can define $f(n+1)$ in terms of $f(n)$ and $n$, for example as $f(n+1)=p((f(n),n))$. Is this correct? If the answer is "yes", I still don't understand the reason.