No, you should never put your photograph on a cover letter. I don’t care how good looking you are, this will make you look extremely unprofessional 100% of the time.
If you include a link to your LinkedIn profile, employers will be able to see your professional headshot here if they really want to see what you look like. If you do include a LinkedIn link, make sure you do have a professional photo and up to date information on your page.
If the application requires a photo, you are best to add it to your resume.
You can use RecruitablehHub professional templates that include space for headshots here https://resume.recruitablehub.com/onboarding