Context-free grammar of the concatenation of a string S and subsequence of reversed S
Let me start with a grammar for the language of words of the form $x\#x^R$:
$$
S \to aSa \mid bSb \mid \#
$$
This is our starting point. Now there are two interpretations of the term subsequence. Under the first interpretation, a subsequence is obtained by removing any number of symbols. This just means that we can omit the second side of each production of the form $S \to \sigma S \sigma$, resulting in the grammar
$$
S \to aSa \mid bSb \mid aS \mid bS \mid \#
$$
Under the second interpretation, the subsequence must be contiguous. Each derivation should therefore be in three steps: first only on the left side, then on both sides, then finally again only on the left side. This requires three nonterminals, and results in the following grammar:
$$
S \to aS \mid bS \mid T \\
T \to aTa \mid bTb \mid U \\
U \to aU \mid bU \mid \#
$$