CFG for $\{a^ib^jc^k \mid i \neq j+k\}$
I would start by finding a CFG that generates the balanced version $\{a^ib^jc^k\mid i=j+k\}$ from nonterminal $C$.


   $C\rightarrow aCc\mid B \\ B\rightarrow aBb\mid\varepsilon$


Now, as you correctly note, there are two possibilities on how to get the unbalanced version; either there is $(a^+)$ to the left, or $(b^+\mid b^*c^+)$ on the right. To deal with the former possibility, you simply add the following rules.

$
S\rightarrow aL\\
L\rightarrow aL\mid C
$

You can deal with the other two possibilities in similar manner.


   $ S\rightarrow R_bb\mid R_cc\\R_c\rightarrow R_cc\mid R_b\mid C\\R_b\rightarrow R_bb\mid B$