How to choose a word to apply the Pumping Lemma?
You are on the right track. You just need to continue observing and trying. The trials and errors you have done might have been done by many experienced users more or less in different ways.

As you have found $w=c^n$ does not work. That word is too simple to be pumped outside of $A$, whether we will pump it down or pump it up.

Let us continue your preference/acumen to ignore the $a^n$ part. Let us choose word $b^mc^k$, where $m&lt;k$. If we can pump substring of $b^m$ enough time, the number of $b$'s in $w$ will become larger than $k$ eventually. To ensure what will be pumped is a substring of $b^m$, we can let $m\ge p$, where $p$ is the pumping length.

So let us try $b^{p}c^{p+1}$. I will let you check that this word, if pumped up, will be no longer in $A$.

However, the number of $b$'s in $w\in A$ must be even. If $p$ is odd, $b^pc^{p+1}$ is not in $A$ in the first place. Well, in that case, we can choose $b^{p+1}c^{p+2}$.

In general, how to choose the right word so that it could pump down or up out of the language might feel like a bit of black magic without much pattern initially. However, there is actually not much  choices involved. You may learn from this answer and this answer.



Exercise. Prove the following language is not regular.
$B=\{a^nb^mc^k\mid n,m,k \in N \wedge n\operatorname{mod}2=1=m\operatorname{mod 2}\ \wedge m&lt;k\}$