Insert-only Levenshtein distance
Your problem appears to be:


Given strings $S,T$, determines whether $S$ appears as a subsequence of $T$.


There are several ways you could approach this.

One way is to use dynamic programming and apply a straightforward modification to computing the Levenshtein edit distance, where you set the cost of an insertion to be 1 and the cost of a deletion or replacement to be $\infty$.

Another solution to use a string-matching algorithm.  Build the regexp $.\star S_1 .\star S_2 . \star \cdots . \star S_n . \star$ and check whether $T$ matches this regexp, using any standard method (e.g., convert the regexp to a NFA or DFA).  For instance, if $S$ is the string HOT and $T$ is the string HELLO THERE, then you will check whether HELLO THERE matches the regexp .*H.*O.*T.*.  I suspect this might be faster, but the best way to find out is to try it and see.