I'm not inclined at this time to look for more information about "pancake on his bum"

But I want to share that a perhaps unrelated line in some children's rhymes is "pimple on his rum tum tum".

Here's an example of a rhyme/song that includes that line:

Subject: RE: Lyr Req: Ching Chung Chinaman Song
From: Fliss - PM
Date: 19 Jul 04 - 02:17 PM

This is what we used to chant in the playground in the 1950's in Shropshire.

Chin Chin Chinaman bought a paper doll
He washed it and dressed it and then it caught a cold
He sent for the doctor
The doctor couldnt come
Because he had a pimple on his rum tum tum.

thread.cfm?threadid=71726&messages=52#1229116

The last line of this rhyme is also given as "Because he had a pimple on his bum bum bum".

Subject: RE: Lyr Req: Ching Chong Chinaman Song
From: GUEST,Steve Gardham - PM
Date: 05 May 08 - 06:42 PM

Similar version to one posted above.
Hull 1950s street rhyme

Chinky Chinky Chinaman bought a penny doll,
Washed it, dressed it, called it Pretty Poll,
Sent for the doctor, doctor couldn't come,
Because he had a pimple on his bum, bum, bum.

-snip-

Also see this very interesting Wikipedia page about the history of taunts against the Chinese in the USA and elsewhere: http://en.wikipedia.org/wiki/User:Andrew_Robertson/names