Finding Distant Primes
JavaScript (ES6), 103 bytes
Expects (m)(k)(n) and returns the \$n\$-th \$(m,k)\$-distant prime (1-indexed).

m=&gt;k=&gt;F=(n,p)=&gt;(g=d=&gt;p%--d?g(d):d^1)(p)|(g=(n,K=1)=&gt;K&gt;n||K&lt;n&amp;k&gt;1&amp;&amp;g(n,K*k))(p+m)*g(p-m)||--n?F(n,-~p):p

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Commented
m =&gt; k =&gt;            // outer functions taking m and k
F = (n, p) =&gt;        // inner recursive function taking n
(                    // and using p = current prime candidate
  g = d =&gt;           // g is a recursive function taking a divisor d
    p % --d ? g(d)   //   decrement d until it divides p
            : d ^ 1  //   return a truthy value if p is composite
)(p) |               // initial call to g with d = p
(                    //
  g = (n,            // g is a recursive function taking n = p + m or p - m
          K = 1) =&gt;  // and K = k ** x
    K &gt; n ||         //   if K is greater than n
    K &lt; n &amp; k &gt; 1 &amp;&amp; //   or K is less than n and k is greater than 1:
      g(n, K * k)    //     do a recursive call with k raised to the next exponent
)(p + m) *           // unless the above function is truthy for p + m ...
g(p - m)             // ... and for p - m (meaning that we don't have |k**x-p|=m)
                     // or p was composite:
|| --n               //   decrement n
?                    // if all of the above is truthy:
  F(n, -~p)          //   try again with p + 1
:                    // else:
  p                  //   success: return p