Can I upgrade macOS HighSierra V10.13.6 27in late 2013?

Your iMac can be updated to macOS 10.15 "Catalina" but no higher.

However, once you go past 10.14 "Mojave" your older apps will not work. See:

32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support

This free tool will help you find what apps will not work with Catalina and check for possible updates.

https://www.stclairsoft.com/Go64/index.html

NOTE: you may have to repurchase some software packages like Office.

Ah! In the last week I have seen several posts from Google Mail users that suggest to me that Google is no longer recognizing older macOS versions. Only they can tell you if Mojave or Catalina will still work with their system.

https://support.google.com/mail/?hl=en#topic=7065107