[verb ます-stem] + 損ねる / 損なう / 損じる [noun] + を + 損ねる / 損なう / 損じる
損ねる, 損なう and 損じる after a ます-stem mean "fail to do": you meant to, or tried to, and it didn't come off. 損ねる and 損なう are close to interchangeable here, and 損ねる is the one you hear most in conversation.
The failure can be a missed chance or a botched attempt. かきそんじる書き損じる and かきそこなう書き損なう are writing something wrongly, and a spoiled sheet is a かきそんじ書き損じ.
損じる is the old, formal member of the three. You meet it mostly in かきそんじる書き損じる and in the proverb below, with しそんじる仕損じる ("to botch"), and rarely elsewhere.
As verbs in their own right, all three take a noun with を and mean "harm", "damage". 損なう goes with health, trust and reputation; 損ねる most often with someone's mood.
These go beyond "fail to" and "harm".