Constantinescu et al. introduced the homogeneous weight on the integer residue ring$\mathbb {Z}_{m}$which can reflect more information compared with the Hamming weight. Few homogeneous weight linear codes over finite chain rings have important applications in cryptography, lattices, modular forms and combinatorics. In this paper, we construct an infinite class of cyclic codes over the finite chain ring$\mathbb {F}_{p^{t}}[\omega]/(\omega ^{2})$by the trace function, and determine their homogeneous weight distributions by applying the theory of exponential sums. In order to investigate the minimality of linear codes over finite chain rings, we firstly present the necessary and sufficient condition for linear codes over the finite chain ring$\mathbb {F}_{p^{t}}[\omega]/(\omega ^{2})$to be minimal or almost minimal by the Hamming weights of codewords. Then, based on the proposed condition and few Hamming weight cyclic codes, we give several classes of minimal and almost minimal linear codes. Furthermore, we derive several families of strongly regular graphs, strongly walk-regular graphs and triple sum sets by few homogeneous weight linear codes.