IRF is generally taken with a colloidal solution; sometimes, solvents or water have some traces of impurities which could give you unwanted peaks. In the IRF, the bump is common. These bumps are called afterpulses. The bumps are caused by regular photoelectrons that are scattered at the first dynode of a PMT or at the front face of the channel plate of an MCP PMT, and re-enter the amplification system with a delay of a few picoseconds or nanosec. The difference compared to true afterpulses is that such scattered photoelectrons are not related to a previously detected photon. They are thus recorded and, because they are delayed, cause said bumps in the IRF.
The decay profile of the sample would also exhibit a bump in the same region due to the influence of the instrumentation. I didn't grasp the concept of subtracting the Instrument Response Function (IRF). If you're suggesting that the IRF was manually subtracted from the data through a normal subtraction process, that would be incorrect. The IRF cannot be manually subtracted from the data. Instead, one can determine the time zero using the IRF data and align the time zero of the sample data accordingly. Then, the data can be fitted with exponential functions. Another approach to handling the data is deconvolving the sample data with the IRF, which is slightly more intricate than fitting exponentials.