There are cases where branding with the word "cheap" might be fine, but they're outnumbered by cases where that isn't the best idea.

If you offer 1 product that is literally identical to what the other people are offering and your entire business model is based around undercutting their prices, then "cheap" might be perfectly appropriate.

For example, if you were booking hotels or flights – the very same hotels and flights as the other guys – but at cheaper prices, then I'd see "cheap" as an important part of your identity.

Without knowing what your service is, it's impossible to give specific advice. Happy to chat for 15 minutes about your branding / domain name options. That's what I do full time.