Both 端 (タン) and 緒 (ショ) are On'yomi. 端緒 means the beginning, start, an initial step, or a clue/lead to something. Incorrect readings like たんちょ use チョ, which is a common Kun'yomi for 緒 but incorrect here. はしお and はしぐち are also Kun'yomi for related kanji or meanings.

端 (タン) も緒 (ショ) も音読みです。物事の始まり、糸口、手がかりを意味します。たんちょは緒の訓読み、または他の音読みを当てた誤りです。はしおやはしぐちは訓読みです。