I am designing a logo for teespring... I have used Microsoft Word for font, then exported word to PDF, which I have then saved on my Sony Xperia. I then take a screenshot of the PDF (as it is open on Adobe Acrobat on my phone). I now have a PNG image that I can save on my computer, crop and use on teespring. Will this PNG image support CMYK and if not, how will this effect the quality of the print?
teespring has quite comprehensive guidance about file formats, this(+its sublinks)
There you can find "PNGs with transparent backgrounds are the preferred file format" and "We use an RGB workflow. Our DTG machines print Red and Green, as well as the traditional CMYK, which allows for a larger color gamut".
Be sure you really understand the requirements Pixel dimensions are essential!
CMYK is out of reach for the people who have only MS Word, PowerPoint, their Open Source counterparts and simple RGB drawing programs. Ignoring RGB = ignoring most of the potential customers => teespring accepts RGB images.
Screenshots, altough easy to do, are not the way to high quality! If you are going to do this more than once, consider to learn some proper image design and processing programs which accept text, drawings and photos + have tools for transparent backgrounds. That's not necessarily the high cost Adobe stuff, pro quality results are possible also with cheaper alternatives (for ex. Serif Affinity series) and freeware (GIMP, Inkscape + others)
PNG and CMYK: PNGs are RGB. No proper image editor allows you to save a CMYK image as PNG. In theory you can make some hack and save CMYK numbers as R, G, B and transparency values, but then you were the only one who possibly could read it right. Open it in any generally available image viewer and the result will be a mess.