Fully Homomorphic Encryption (FHE) allows arbitrary computation on encrypted data, enabling a privacy-preserving machine learning as a service (MLaaS) model in which clients encrypt their sensitive inputs and remote servers perform inference without ever seeing the plaintext. Despite this promise, the computational cost of encrypted inference remains a major obstacle, especially when neural networks are trained in plaintext but evaluated on encrypted inputs. We present an acceleration framework for privacy-preserving neural network inference using gate-based FHE. In our scheme, the client encrypts its input and transmits it to a server that holds a pre-trained plaintext neural network. By carefully exploiting constant-aware arithmetic for the dominant ciphertext \(\times \) plaintext weight multiplications, we transform each multiplication into a small number of additions, subtractions and inexpensive shifts via non-adjacent form (NAF) recoding and bit-slicing. This drastically reduces the number of costly bootstrapping operations. Experiments on a suite of 23,900 ciphertext \(\times \) constant benchmarks and convolutional layers demonstrate that our method achieves a \(4.23\times \) average runtime reduction compared to standard FHE compilers, reducing bootstrapped gate counts by up to 65% and accelerating encrypted CNN inference by nearly \(3 \times \) . These results show that FHE, when coupled with constant-aware compiler optimizations, can support practical, privacy-preserving neural network inference in real MLaaS scenarios.