Smallest Hamming distance to a palindrome containing a substring
Pyth, 19 bytes

hSmsnVQd/#z_I#^+Qzl


Demonstration

Extreme brute force approach. Generate all strings of the appropriate length with characters in either string, filter for palindromes and for containing the second input, map to hamming distance from first string, output smallest.

Explanation:

hSmsnVQd/#z_I#^+Qzl
hSmsnVQd/#z_I#^+QzlQ     Variable introduction
                         Q = string A, z = string B.
               +Qz       Concatenate A and B
              ^   lQ     Form every string of length equal to len(A)using
                         characters from the concatenation.
             #           Filter on
           _I            Invariance under reversal (palindrome)
         #               Filter on
        / z              Nonzero occurences of B
  m                      Map to
    nV                   !=, vectorized over
      Qd                 A and the map input
   s                     Sum (gives the hamming weight)
hS                       Min