He 'tempered' his criticism 'with' a few words of encouragement. = He 'tempered' his criticism 'by' adding a few words of encouragement.
He was upset but 'kept his temper'. [=remained calm; did not become angry, begin shouting, etc.]
Higher interest rates have 'tempered' [='lessened'] the demand for new houses.
(..) and he wouldn't have a lunatic living in his house. He worked out his frustration with life on the creek every morning. And in the evenings, to temper the sting of loneliness, there was Martha Shaw. Martha was a war widow who lived in Quail Ridge, a town away. So you want to go out (..)2004 The Notebook