In Modim in the Shmoneh Esre we say המרחם כי לא תמו חסדך the question is whether the מ in המרחם should have a daggesh or not, and thus what sort of Shva should be under said מ?
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It should not have a dagesh. Throughout Tanach, most every mem with a sh'va as the first letter of present-tense verb in piel or pual has no dagesh after a he hay'dia according to the m'sora (and the same is usually true of any mem with a sh'va), and I have no reason to believe that that changed in later Hebrew (certainly not by the time "Modim" was written).
That said, I'm not sure what kind of sh'va the mem has. At least possibly, the sh'va is na anyway.
For many words in Tanakh that begin with he hayedia followed by mem-shewa, even though the mem lacks daghesh, the Masora prescribes shewa naʿ (mobile shewa), indicated by gaʿya on the preceding pathah. We should not at all assume that the words of the siddur were pronounced as they were in the Tanakh. The language (and pronunciation) of Hazal often differed. The preponderence of evidence suggests that unless one is carefully and holistically following an established authentic reading tradition in which the shewa is nah, the shewa should be considered naʿ.