Visible-light communication (VLC) stands as a promising component of the future communication network by providing high-capacity, low-latency, and high-security wireless communication. Superluminescent diode (SLD) is proposed as a new light emitter in the VLC system due to its properties of droop-free emission, high optical power density, and low speckle-noise. In this paper, we analyze a VLC system based on SLD, demonstrating effective implementation of carrierless amplitude and phase modulation (CAP). We create a low-complexity memory-polynomial-aided neural network (MPANN) to replace the traditional finite impulse response (FIR) post-equalization filters of CAP, leading to significant mitigation of the linear and nonlinear distortion of the VLC channel. The MPANN shows a gain in Q factor of up to 2.7 dB higher than other equalizers, and more than four times lower complexity than a standard deep neural network (DNN), hence, the proposed MPANN opens a pathway for the next generation of robust and efficient neural network equalizers in VLC. We experimentally demonstrate a proof-of-concept 2.95-Gbit/s transmission using MPANN-aided CAP with 16-quadrature amplitude modulation (16-QAM) through a 30-cm channel based on the 442-nm blue SLD emitter.