In your Sprout Social inbox, you can use Brand Keywords to pull in social messages that you want to reactively manage and respond to. See more on Brand Keywords here.
Include search operators in your Brand Keywords to help filter out irrelevant messages and zero in on the keyword matches that are important to your business. To make sure you don't see messages from a specific Twitter user, you can use the - operator to remove any of their results from your keyword results. For example:
This brand keyword/search operator string example would pull in messages from Twitter that contain the word Sprout, as long as they weren't from the handle @SproutGKeenan: