No you don't drop the "Y". The "Y" sound is there. Hope this helped.

I agree with Miss Alexis and Su.Ki. (I think I heard a very quick /j/ in Su.Ki.’s recording, but I certainly wouldn’t count on it when listening to fast speech.) After a postalveolar consonant, the y (/j/) often gets slighted or dropped in natural relaxed speech. This is a corollary of the blended pronunciation rules, where /dj/ is pronounced /ʤ/ and /sj/ is pronounced /ʃ/. Basically, the postalveolar consonants of English are analyzed as containing the /j/ phoneme. When it happens within a word (which is all the time), the dictionary transcription reflects this. (Sadly, dictionaries don’t show pronunciation across word boundaries, however.) So Cambridge shows “gradual” is pronounced /ˈɡrædʒ·u·əl/ (not /ˈɡræd·ju·əl/) and “sure” is pronounced UK ​/ʃɔːr/ US ​/ʃʊr/ (not with an initial /s/).  

Adrian: I can’t speak for Alexis, but I most definitely teach my students how native speakers really pronounce English most of the time. (Since this is unconscious, it is often quite different from the way native speakers believe they speak.) Not only does this training make my students sound more natural, but most importantly, it allows them to understand real (educated native speakers' natural) English much faster than would otherwise be possible.

I think that those transcriptions look like a fairly accurate reflection of natural relaxed speech.

Hi Benny!

In connected and quick speech, the 'y' sound (I believe that is the [j] semivowel] be deleted (at least in American English).

If you are enunciating every word fully, however, you will hear that [j] before the schwa (upside down 'e').

Personally, I drop that [j] when speaking normally. It is only with precise pronunciation that you will hear it from me. :-]

@Miss Alexis Thanks so much. I really appreciate it.

@Adrian Thanks for answering.

I just want to know if the elision of y or /j/ exists across the word boundary.

Because it's quite common within the word like /njuːz/ in British English that becomes /nuːz/ in American English.