What is the correct pronunciation HaGeshem or HaGoshem?
Since geshem changing to gashem is dependent on whether the word comes at the completion of a phrase (esnachta or sof pasuk), the real question is:
Is the ability to make rain a praise in itself (ending the phrase- hagashem), or is it only a praise if it is used to sustain life and must be followed by mechalkel chaim b'chesed (hageshem)
There are many in each camp. Interestingly, even though Rav Moshe writes hagashem, the sefer Tefila K'hilchasa quote a pamphlet (Mechalkel Chaim) which claims much support for hageshem and claims that Rav Moshe retracted.
Actually the sefer you are refering to is Birchos HaChaim. It is available on Otzar HaChochma. Based on what I have read so far, the correct reading would be haGESHEM. (I have not found yet where he says that Rav Moshe retracted, but I have only gone through half of the 97 pages). The nusach of haGASHEM seems to have been put in by a maskil... He also discusses if haTal should be with a kametz or a patach. Check the sefer for a full explanation.
See excellent article here: http://ohr.edu/this_week/insights_into_halacha/4903. it explains not only the why's but also the who's.
another relevant link : http://www.dinonline.org/2011/11/07/pronunciation-of-gashemgeshem/
also i would not rely lmaaseh on the tefillah kehilchasa's claims that rav moshe retracted. if he would have, it would be listed in more reliable sources. many others including rav elyashiv said gashem as well. I think he is confusing it with rav shlomo zalman auerbach who is quoted in halichos shlomo as retracting after reading birchos chaim. the tefillah khilchasa also claims that the gr"a said geshem - and that the steipler said gashem - both the exact opposite of every other source.