In the December 2012 (inaugural!) edition of Anahorish Bungaku — which is indeed named after the Seamus Heaney poem — Nishizawa Kazumitsu 西澤一光 has an article entitled "Tekisuto to shite no Man'yōshū" テキストとしての『万葉集』 ("The Man'yōshū as text") where he mentions that:

In fact, innumerable quotations from Chinese texts are woven into their [the Man'yōshū poets'] original text[s]. For example, Kakimoto no Hitomaru's orthographic expression (文字表現) indicates not only that he was an avid reader of the Classic of Poetry and the Selections from Literature, but also that at the same time he had read the Mahāyāna Mahāparinirvāṇa Sūtra and the Suvarṇaprabhāsa-sūtra closely enough to recite [parts of] them from memory.

This sounded interesting, so I tracked down the reference, which was to another paper of Nishizawa's: "Man'yōshū to 'mujō'" 『万葉集』と「無常」 ("The Man'yōshū and 'impermanence'"), in Takaoka-shi Man'yō Rekishikan ronshū 13: Sei no Man'yōshū 高岡市万葉歴史館論集１３　生の万葉集 ("Takaoka Manyou Historical Museum essay collection 13: The Man'yōshū and life"; Kasama Shoten 2010). Unfortunately, though, the evidence for all this sutra-memorizing is thinner than I had hoped.

There is certainly interesting information in the paper, like a brief discussion of Yamanoue no Okura's consistent use of the spelling 世間 (Sino-Buddhist jargon for our impermanent world, corresponding to Pali/Sanskrit loka) for the native Japanese word yo no naka ("[in] the world"). But then the fact that Hitomaro didn't use 世間 for that world — he used 世中 instead — is interpreted as an intentional avoidance of Buddhist jargon on Hitomaro's part, because "it is surely impossible to imagine that [he] ... did not know the Buddhist word 世間". This is a rickety platform to raise a thesis on.

The next piece of evidence is a poem by Hitomaro written 水上如數書吾命妹相受日鶴鴨. This is traditionally read something like:

midu no pe ni/ kazu kaku gotoki/ wa ga inoti/ imwo ni apamu to/ ukepituru kamo

Meaning something like "My life is like marking a count on water [i.e. futile]; How I have prayed to see my love" or "I have prayed my life away to see my love, as futilely as marking a count on water," etc., depending on where/how you cut and some other technical details. Nishizawa notes that 畫水 ("write on water") is a common metaphor in the Chinese canon for futility, and that Keichū pointed out a similar passage in the Mahāyāna Mahāparinirvāṇa Sūtra:

T0374_.12.0367b06: ... 是身無常念念不
T0374_.12.0367b07: 住。猶如電光暴水幻炎。亦如畫水隨畫隨合。

This body is impermanent; it does not abide for an instant. It is like lightning, a storm, illusion and flame; and it is like drawing [a line] in water: no sooner does one part the water than it is reunited again.

Again, though, this is slim pickings. I'll buy that Hitomaro was struck by the metaphor of writing on water when he encountered it in Buddhist writing, and it certainly seems possible that 水上如數書 is a direct reference to 如畫水, but I was hoping for a rather smokier gun than "they share a metaphor that appears in multiple Buddhist texts and that Beaumont and Fletcher also came up with independently in England."