What is impulse response in simple AWGN channel?
A linear system cannot add anything new to its input signal, so additive noise is modeled separately from any linear distortion of the signal. In your case with just scaling and delay, the channel impulse response is $h(t)=k\delta(t-\Delta t)$, and the complete channel is modeled by first filtering the input signal with an LTI system with impulse response $h(t)$ and then adding the noise.