I recently had an experience with a client that was a bit messy, and now they don't want to pay for the work I did. 

They needed some presentation decks, one of which needed to be editable (text to go on top of graphics) in PowerPoint. I told them that I do not work in PowerPoint as a graphic designer, what I can do is create images that can serve as a background template onto which text boxes on PowerPoint can be created. 

From there, my time was highly mismanaged. The person kept directing me to figure out different ways of working around what I proposed and seemed resistant to it. All the research I did, and later confirmed with graphic designers- told me that this was the way to go from Adobe to PowerPoint. I initially did not want to invoice for this time, but they directed me to look into things after I told them I didn't work in PowerPoint. 

They fired after I spent days on end working and researching to appease the woman and see if I could find other ways around around her reticence about working with the approach I outlined about.

They then sent me an email that my work was incomplete and unusable and would not pay me (p.s. they fired me before they even reviewed all of it, and what was reviewed was never stated to be incomplete or un-useable). I explained to them that my time was 1) mismanaged and redirected to research and this delayed design time and 2) that I did everything I was directed to and they fired me before any revisions or edits could be made. If they needed something to be created in PowerPoint from the start, they should have told me. 

So I was hoping to get advice- it is my understanding that designers don't work in PowerPoint and that I worked around this in the way that it seems everyone I talked to would. Can they do this? We agreed to freelancing per hour, not per project, I did not have a contract unfortunately.