Is this hand cipher any more secure than the Vigenère cipher?
Given that the permutation is fixed and the key step is independent of the permutation you can reduce this to an ordinary text-substitution cipher. 

If the key is as long as the input you have a weak one-time pad, because the per letter change is limited to 10 instead of 26.

However if the key is short then you have a Vigenère cipher (if you "decode" with $key=0$ then it is exactly a Vigenère cipher).

In essence you use 2 transformations: a character scramble ($P$  $P'$) and a "xor" ($X+$ and $X-$).

Your algorithm is $text1=X+(P(text_0),key)$for each phase. 

However this is equivalent to $text1= P(X+(text_0,P'(key)))$. 

Then the next fase is $text_2= P(X+(P(X+(text_0,P'(key))),P'(key)))$ which can be rewritten to $text2 = P(P(X+(X+(text0,P'(key) ), P'(P'(key))))$ and so on.

Given that P is easily reversible you can extract $X+(text, X+(P'(key), P'(P'(key)) )$.