What kind of soundcard are you using to record? When connecting from your mixer to the sound card, make sure you are using the soundcard's line input and not the mic input. Can you tell if you are getting digital or analog distortion?
Perhaps you need to set and monitor your input levels more carefully. First set the recording level in the mixer to make sure you're not distorting there. Second, go into the mixer app that came with your soundcard and monitor the input level there. Sing or play into the mic at a typical level and make sure you're not peaking the level. That is, make sure you never go over 0dB. If you do, you'll get digital distortion. (I normally set the sound card input fully up, and if that's too high, reduce the mixer output level.) Finally, go into Cubase and enable the LED monitor for the record channel so you can monitor while recording. This level should look the same as in the soundcard app since Cubase has no adjustment for the input level.