Writing lit review for publication in a good journal: what to do about papers published in dubious journals?

Question

I'm publishing in a field related to applied linguistics, and I'm making use of a new theoretical framework which has been used only in 4 published articles. 1 of those articles was published in an in-house university journal and the other in a very dubious open access journal, i.e. the kind of journals that publish basically anything.

Do I cite and probably critique these 2 papers in my lit review or ignore them?

I've certainly made use of these kinds of journals or conference proceedings in the past myself, which tend to have limited or no peer review. Those older papers of mine haven't really been cited by others, nor would I expected them to have been. But now I'm a bit further along in my career and publishing in journals with good citescores.

The articles aren't terrible, but somewhat misuse/misunderstand the theoretical framework I'm using. One in particular describes and tries to use the framework as if it's quantiative when it's actually qualitiative. I came across the papers late in the writing process, so I can't say they have contributed to my theoretical framework or analysis. I feel if cite them at all I'm giving them weight and weakening the impression readers would have of the framework. One of my main purposes in writing my article is to popularize the framework. I also don't think it's great to cite some obscure paper in a well-known journal just to tear it down.

What does everyone think? Is it better to ignore these papers, or do I have a responsibly to cite and critique all studies which make use of the theoretical framework, however suspect the publication source is? Obviously I'm leaning to the former, but I'm open to a change of mind.

Answer 1

title: lit review for publication

You title seems to suggest your paper is a review paper. However, reading through your question, I understand you're indicating the lit review section of your manuscript.

If your manuscript is a review, you ought to review all four papers that have worked/used the framework.
In this instance, it's the lit review section within your paper, you may, you may not. However, you'll be doing a disservice to the research community by ignoring willy-nilly

One of my main purposes in writing my article is to popularize the framework

In popularising the framework, you'll do well in highlighting/showcasing the ways it's been used and how it should not be used, also how it's been used innovatively.

somewhat misuse/misunderstand the theoretical framework I'm using. One in particular describes and tries to use the framework as if it's quantiative when it's actually qualitiative

I'm not a mixed-methods person and hence don't get caught up in the quantitative and qualitative discourse. I'm inclined towards plurality (multimethods). Without a sight of the paper, I'm not in a position to comment appropriately.
However, it might be that the authors might be attempting quantitising the qualitative framework. If it's not, you used be able to point this out in your manuscript without necessarily 'tearing their paper down'. If it is so, them write on 'innovativeness' of their quantitising approach and how best to use the framework qualitatively.

What does everyone think? Is it better to ignore these papers, or do I have a responsibly to cite and critique all studies which make use of the theoretical framework

Ignoring might not be the best. More so, you've said that the articles in themselves ain't overly 'bad'.
Yes, you should rather critique all (relevant) previous studies. What you should guard against is criticise.