I was far from being able to solve the puzzle, so I decided to check back with the quizmaster, asking for a “lifeline” of sorts. I wanted to show him the full list with the most likely candidates marked. So … here goes (with my selections in green):
1. Fragment 1: ながきよの (nagakiyono)
名が (naga: name is)
流 (naga(su): to drain, pour)
長 (naga(i): long)
長木 (nagaki: long tree!) long + tree
永木 (nagaki: eternal tree?!) eternal + tree
餓鬼 (gaki: (1) brats, kids; (2) (hungry ghost); ghoul) hungry + ghost
額 (ga(ku): (1) picture (framed); (2) amount or sum (of money))
学 (gaku: learning; scholarship; erudition; knowledge)
寄与 (kiyo: contribution; service) unusual + to give
毀誉 (kiyo: disapproval by some and approval by others)
to disapprove + to approve
清 (kiyo(i): clear; pure; noble)
Fragment 2: とうのねむりの (tōnonemurino)
当の (tōno: the … in question)
眠り (nemuri: sleep)
Fragment 3: みなめざめ (minamezame)
皆 (mina: all; everyone)
御名 (mina: name of God (esp. in Christian contexts); name of Jesus) honorable + name
目覚め (mezame: waking) eyes + to wake up
Fragment 4: なみのりふねの (naminorifuneno)
波乗り (naminori: surfing) wave + to ride
船 (fune: boat)
Fragment 5: おとの (otono)
音 (oto: sound, noise)
Fragment 6: よきかな (yokikana)
良き (yoki: goodness; good)
予期 (yoki: expectation, anticipation, hope) in advance + to expect
かな (kana: maybe)