Yes, there are various surviving letters written by Henry VIII and signed by him. See http://www.nationalarchives.gov.uk/henryviii/parchment/writing.htm for an image you can magnify. Others can be found at https://www.theanneboleynfiles.com/7598/henry-viiis-love-letters-to-anne-boleyn/ (click on the photo to enlarge) and http://www.bl.uk/onlinegallery/onlineex/henryviii/greatmatter/letter/largeimage93125.html. You'll also find other in archives or by doing a Google image search.

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