When Jayce buys essence reaver it looks like the CDR on hit only applies to the abilities that are currently available to cast. This means that his gate (E) does not get any of the on hit CDR while in cannon form.
I just wanted to make this post to report in case it was a bug. If not that's fine as I think Jayce will get along just fine without all his Abilities getting reduced. If anything it might force Jayce to stick to range form in order to spam shock blasts, which isn't exactly why I play Jayce, but is still powerful.
I was able to reproduce this 100% over ~5 different attempt where I was paying close attention.