Replacing non-polynomial functions (e.g., non-linear activation functions such as ReLU) in a neural network with their polynomial approximations is a standard practice in privacy-preserving machine learning. The resulting neural network, called polynomial approximation of neural network (PANN) in this paper, is compatible with advanced cryptosystems to enable privacy-preserving model inference. Using ``highly precise'' approximation, state-of-the-art PANN offers similar inference accuracy as the underlying backbone model. However, little is known about the effect of approximation, and existing literature often determined the required approximation precision empirically. In this paper, we initiate the investigation of PANN as a standalone object. Specifically, our contribution is two-fold. Firstly, we provide an explanation on the effect of approximate error in PANN. In particular, we discovered that (1) PANN is susceptible to some type of perturbations; and (2) weight regularisation significantly reduces PANN's accuracy. We support our explanation with experiments. Secondly, based on the insights from our investigations, we propose solutions to increase inference accuracy for PANN. Experiments showed that combination of our solutions is very effective: at the same precision, our PANN is 10% to 50% more accurate than state-of-the-arts; and at the same accuracy, our PANN only requires a precision of 2^{-9} while state-of-the-art solution requires a precision of 2^{-12} using the ResNet-20 model on CIFAR-10 dataset.