Should I cite a really bad research paper published in my small research area? I am working on a special topic in applied math. We are trying to explore a new and largely unexplored area in our line of research. Our hope is to be the first researchers to have solid results in this new line of research.

Couple days ago, my co-author and I have stumbled upon a paper that is titled almost exactly as the topic that we wish to explore and publish. 

However, upon further reading, we realized that the paper (despite being published in a journal) is actually very low quality. Not only that the theoretical guarantees does not support any of the applications in the paper, the theories themselves are also very poorly explained, definitions are missing, handwavy at times, and we suspect that the proofs are wrong, but cannot be verified due to author's lengthy and unconventional proof style, which involves some vague interpretation of results contained in other references, one of which is written in a language that is completely foreign to us and couldn't be found online after a quick Google search.  

This isn't to say that the authors are cranks (even though the paper reads like one). The paper is roughly 10 pages. Properly formatted in the style of the journal. With ample (albeit, strange) diagrams. It is just that the results are confounding, suspicious, and paper is lacking in rigor, despite being peer reviewed and published in a journal. 

For what it is worth, the authors are not working at a top-tier research institute and the journal is not the highest quality of this field.

What should we do when we start our writing process?


Should we painstakingly go through the process of providing a
critique of a paper that is written in such a way that it is difficult to critique (without dismissing it as poor writing)? 
Or should we ignore the paper and pretend that we didn't see it? (Despite the fact we wish to work on the same, very specific, topic?)
Ultimately, what should you do when you see bad research papers like these?
Yes you should cite it because:


You know about that paper
You read it in the context of preparing your work. Finding gaps and flaws in previous works is a major part of designing and refining your own research 
By your own judgement it’s highly relevant.


Explain your rebuttal of their claims or method using proper scientific arguments. If I were you I would refrain from using words like “hand waving” or “cranks” in a scientific context.

Whether or not you are the first to discover something isn’t really something you can control. It’s possible that they actually did find something of value but were bad at communicating it. From what you say there seem to be still time for you to be the first to get it right but you won’t achieve that by ignoring previous work.