ついでに
on the way, while one's at it
You are going out anyway, so the dry cleaning can go along with you. The same つい as つい and ついに, except that this one is riding on something else. Not a trip, an extra stop on one: ついでに.
途中 とちゅう is partway along a journey. ついでに is doing a second thing because you were going regardless.
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Emaki is not affiliated with, or endorsed by, Kaishi 1.5k or its authors. The word, reading and meaning come from the Kaishi 1.5k deck, used with their permission. The mnemonic, the note and the kanji breakdown are original work, published under CC BY-SA 4.0.