So-called liberals often have a troubled relationship with the Prayer of Humble Access, while Catholics may resent its temporary ascendancy over the Agnus Dei. It has been derided as the "Prayer of Humble Excess" and the "Humble Crumble Mumble," but Prayer Book champions wear it as a badge of pride.
In the ordo missae on page 230 of the Book of Alternative Services (that is, the Canadian Rite I, if you will) the Prayer is retained in a redacted form. The obvious source of consternation, "we are not worthy so much as to gather up the crumbs under thy table", is still there. ("Even though the dogs are!" as one ordinand of my acquaintance remarked). Gone, however, is the less evidently objectionable "that our sinful bodies..." In Rites for a New Age, Michael Ingham suggests that this is to avoid the suggestion that the elements can offer remission of sin. This seems dodgy to me, as I think it perfectly orthodox to understand receiving Holy Communion with faith as being unto forgiveness of sin. As a concession to comprehension of Puritanism, the clause "in these holy mysteries" had of course been given up before the first Canadian revision of the prayer book.
It is this understanding of the sanctifying nature of the Eucharist that causes the "crumbs" passage to grate. The Roman Catholic Church, for instance, teaches that venial sin is absolved by reception of the Sacrament. While it is faithful to the Pauline understanding of the Eucharist, I think we have to question how well that tradition squares with that of Jesus himself. Are we to think of Holy Communion as a reward for sanctity or a means thereof? I think that the latter is perhaps a more sensible approach. Since Our Lord commanded us to "do this" it seems disingenuous to plead our unworthiness routinely before going on to fulfil the command.
So there are Catholic grounds for suspicion. There also the "liberal" grounds. They too have a point: it seems liturgically problematic at best to have such an abject prayer after the Confession and Absolution. It might be possible to incorporate such language into the penitential rites if desired, but it seems out of place in the Communion Rite, for how have we made it this far if not having been counted worthy?
Personally, I'm very much a "two book" Anglican. There was a move a few years back to try to get an American-style single book of two rites. I think that both BCP and BAS have their own integrity however. The BCP reminds of the truth of our fallen nature, our inability to save ourselves, and our reliance on God's mercy. The BAS reminds of the truth of redemption by the true Paschal lamb. I wouldn't want us to lose either of these traditions we have inherited. (In some American parishes, Rite I is used in Advent and Lent). The Book of Alternative Services restores practices such as the imposition of ashes, chrismation, Holy Week rites, and an Epiclesis. The rubrics allow for the interpolation of minor propers in the appropriate places. The BCP is dense with wonderful prayers that should be preserved intact.