ersehnt
long-awaited
Intensely desired over a long period of time; eagerly awaited.
Nach der langen Dürre brachte der ersehnte Regen endlich Erleichterung.
After the long drought, the long-awaited rain finally brought relief.

long-awaited
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long-awaited
Intensely desired over a long period of time; eagerly awaited.
Nach der langen Dürre brachte der ersehnte Regen endlich Erleichterung.
After the long drought, the long-awaited rain finally brought relief.

'Ersehnt' (long-awaited) is the past participle of the verb 'ersehnen' (to long for). This verb consists of the stem 'sehnen' ('sich nach etwas sehnen' = to long for something) and the prefix 'er-', which here emphasizes reaching a goal.
Der ersehnte Frieden ließ lange auf sich warten.
The long-awaited peace was a long time coming.
This adjective is almost always used before a noun (attributively). Using it after the noun as part of the predicate is rare and often sounds poetic.
Endlich kam die ersehnte Zusage für den Studienplatz.
Finally, the long-awaited acceptance for the university place arrived.
Common nouns combined with 'ersehnt' (long-awaited) are positive and long-awaited things such as 'der ersehnte Sieg' (the long-awaited victory), 'die ersehnte Pause' (the long-awaited break), 'der ersehnte Anruf' (the long-awaited call), or 'die ersehnte Nachricht' (the long-awaited message).
Nach der anstrengenden Arbeitswoche genoss er die ersehnte Ruhe.
After the exhausting workweek, he enjoyed the long-awaited peace and quiet.
The word 'ersehnt' (long-awaited) belongs to an elevated, slightly poetic style. It expresses very strong and prolonged waiting.
Der ersehnte Moment des Wiedersehens war sehr emotional.
The long-awaited moment of reunion was very emotional.
expected
'erwartet' and 'ersehnt' can both have to do with something that is still to come. The difference is the feeling behind it. 'erwartet' is more neutral: you expect an outcome or an event. It does not have to be especially pleasant or important. 'ersehnt' is much more emotional: you strongly want something, often for a long time. The words can be confused because something that is 'ersehnt' may also be expected. But not everything expected is 'ersehnt'; an expected bill, for example, is not necessarily wanted.

Nach vielen Monaten begann endlich der ersehnte Urlaub.
After many months, the long-awaited vacation finally began.
Die erwartete E-Mail kam am Nachmittag endlich an.
The expected email finally arrived in the afternoon.