励む and 励ます: working at an activity and encouraging someone
Compare effort in pronunciation or competition practice with encouragement of a friend or runner. Identify the activity with に, the recipient with を, and the crowd's cheers as the subject.
Working hard at one's own practice
励む (はげむ) includes working hard and devoting oneself. 「彼女は毎朝、発音の練習に励んでいる。」 means “She works hard on pronunciation practice every morning.” She is the actor, and pronunciation practice is the activity linked by に.
「大会を目指して日々稽古に励む。」 means “I practice diligently every day with the competition as my goal.” This describes the person's own practice directed toward a competition. It does not become a scene of verbally encouraging someone else, and the goal does not supply a competition result.
Encouraging a friend or a runner
励ます (はげます) means to encourage. 「落ち込んだ友人を言葉で励ました。」 means “I encouraged my dejected friend with a few words.” The friend marked by を receives the encouragement, and words are the means.
「観客の声援が走者を励ます。」 means “The crowd's cheers encourage the runner.” The cheers are the subject, and the runner is the recipient. This relationship does not describe one spectator working hard at their own practice.
Track the activity, actor, and recipient
For 励む, identify who works at which activity; for 励ます, identify what or who encourages which recipient. The direction differs between one's effort and encouragement directed toward someone. The forms alone do not supply a rule for automatically generating one verb from the other.
In the competition example, 大会を目指して gives the goal, while 稽古に names the activity connected to 励む. 発音の練習に fills that activity position in the pronunciation example. In the crowd example, が marks the cheers as the subject and を marks the runner as the recipient of encouragement. Read を with its verb: 大会 is the goal expressed by 目指して, while 走者 receives the encouragement expressed by 励ます.
In 「観客の声援が走者を励ます。」, what is the subject and who receives encouragement, and how does that direction differ from 励む in practice?
Show the answer and why
The crowd's cheers are the subject, and the runner receives encouragement. In a practice sentence with 励む, a person works at their own activity. Encouragement directed toward the runner does not mean that the spectators themselves practised.