What does "please with sugar and knobs on" mean?

It's three different "intensifiers" used together, effectively "Please please please please!"

Pretty as in that was pretty stupid is quite common in the slightly childish/winsome/wheedling entreaty Pretty please?.

With [brass] knobs on is a well-established intensifier (cf with bells on). I've not come across with sugar used in this way, and certainly the conjunction of all three isn't what I'd call a standard idiom, but in context the meaning is obvious.

EDIT: Note that "The same to you with [brass] knobs on!" is (rather dated) British slang, which explains why many people aren't familiar with it. Outside of that specific retort, with knobs on was never particularly common, so for any one under 50 (or not British) I've obviously overstated the case by saying it's a "well-established intensifier". Nevertheless, it is just an intensifier, with no other connection to "pretty" and "with sugar on".

"With sugar on," indicates you are offering something desirable or valuable as an incentive.

"With knobs on," comes from 1920's slang, and meant that something was embellished, for example an iron bedstead with knobs on becomes that bit more special.

The two phrases together are just meant to accentuate the request, as is the repetition of "pretty," - the request becomes special, different, more important.

See also "Pretty please with sugar on top" .

As you know please is a modifier for making a request polite, and by implication making it more likely to be fulfilled. A child, for example, might take this "magic" quality of please and try to dress up the phrase. Pretty please indicates more urgent pleading. Pretty please with cherries on top is rather abject. Anything desirable "topping" could be suggested here, as one would add to ice cream or pancakes and the like: sugar, syrup, or in this case knobs probably meaning a knob of butter.

"Knobs" is not a common measurement in American English; usually it is a vulgarism for breasts or nipples.

I've heard pretty please with sugar on top since I was a youngster. It's is a fairly common (and somewhat childish) way of pleading with someone:

I've never heard the "with sugar and knobs" variation, not until today. Based on a quick Google search, though, it appears to be nothing more than a simple variant on the "sweet topping" motif; knobs apparently refers to gobs of butter:

How much butter is in a knob of butter? That's a question for a different StackExchange.