“You should always submit a cover letter, even if they don't ask for one.”
You're sending an email to a younger person who's applying for her first job out of college. She asked you if you think it's important to include a cover letter when she applies for jobs. You do think it's important, so this is what you write.
You should always submit a cover letter, even if they don't ask for one.
When you "submit" an application, a homework assignment, or some other official document, it means that you give it to the person or organization who has asked for it and who has authority over it. For example, you "submit" a job application to the company that might hire you.
The word "submit" is a slightly formal word. In casual conversation, you can use the phrase "turn in (something) or "turn it in":
Make sure to turn in your application by Friday.
In the U.S., a "cover letter" is a letter that you include with your resume when applying for a job. It's usually a few sentences long and describes why you are applying for the job and why the company should consider you for the position.