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.
(..) It is also a lie. This memory has been tempered with by the same person whose memory it is. Our old friend, Professor Slughorn. But why would he temper with his own memory? I suspect he's ashamed of it. Why? Why, indeed. (..)2009 Harry Potter and the Half-Blood Prince