To whom it may concern (if you can find a name through LinkedIn or the company’s website, even better!),
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I appreciate interns can sometimes be difficult to fit into a busy team and aren’t always considered a priority.   
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If offered an opportunity I would be as useful and focused as I could be when needed and happy to read and learn when the team do not have time to involve me.  I am great at researching and note-taking and would willingly turn my hand to any task in return for the opportunity of learning about your business and its operations.

I am committed to pursuing a career in X industry and your company stands out as an exciting employer within that environment.
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I have learned much throughout my (list course/education/experience), I have (list accomplishments/evidence passion for the role, ie, I have recently taken part in Springpod's Law virtual work experience where I particularly enjoyed…).

I understand that you have a formal recruitment process, but if you are currently experiencing a high workflow then I’d like to be of assistance. My CV is attached.

I am available to start from (date) and would appreciate any opportunity within your team.
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Thank you for your time and I look forward to hearing from you.

Completing some Virtual Work Experience, or Subject Spotlights will strengthen your CV even more! And make finding some in person experience even easier!

Springpod's simple guide to applying for internships and in-person work experience. Step by step guide on how to get work experience and easy template to use for applications

Work Experience/Internship email template