The person who corrected you was wrong. Removing mentale substantially changes the meaning of your sentence.

Maybe the reason why (s)he did it is santé mentale seems odd here. In my experience, it is not used to describe the activity sector but people’s health, e.g. :

Je m’inquiète pour la santé mentale de ton ami.

When referring to a job, I would say instead:

Je recherche (encore/toujours) un travail en psychiatrie.

or, depending on exactly what you are looking for, which you fail to state :

Je recherche (encore/toujours) un travail dans le secteur de la psychologie, de l’assistance psychologique, etc.

j’y cherche encore is not correct in this context. Y cannot replace santé mentale, because it is not a place name. You have to repeat santé mentale or replace it with another noun.

Je cherche encore un travail dans ce domaine (or dans ce secteur).

Concerning the fist part of your question I would say you were right to repeat santé mentale. We do not have the entire conversation but you could have changed your mind in between the two times you talked about it. And the difference is the same in French between santé mentale and santé.

Edit to answer the OP’s comment.

Let’s imagine you were asked :

Est-ce que vous cherchez encore du travail ?

And you wanted to replace du travail by a pronoun, you’d have to use en.

Oui, j’en cherche toujours (encore) dans ce secteur.

You can look at the answers to this question on French Language:
Quelle est la différence entre les pronoms « en » et « y » ?