I have seen this a handful of times in my fifteen years in the business. The procedure for removing the "partial foreskin" is obviously quite different than a run of the mill circumcision - for what I hope are obvious reasons.
When there is a partial foreskin, the circumcision "cleans up" the extra skin that is preventing the circumcised look. While not a full circumcision, it is nonetheless a surgical procedure and should be treated the same way a circumcision is, insofar as that if the baby is ill, we would delay the bris 7 days until he is completely healthy.
But when there is no foreskin at all, and the child only needs "Hatafat Dam," which is a (mostly symbolic) minimal drawing of blood, will the same rule of waiting apply?
Here is from "Bris Avos" - same chapter (see previous blog post), paragraph 4.
In the Responsa "She'eilat Shalom" the question was asked whether a child born without a developed foreskin ("Mahul") [who cannot be circumcised, but would need "hatafat dam"] can have the hatafat dam on the 8th day if he became ill. [Hatafat Dam is a ritual which does not require follow up medical care]. The answer he gave is that it is not only forbidden to do anything to him while he is ill, but one would need to wait the same requisite 7 days after he is healed in order to do the Hatafat Dam, just as one would wait the requisite seven days before doing a full circumcision.