Let $\mathbb{F}_{p^m}$ be the finite field of order $p^m$, where $p$ is an odd prime and $m$ is a positive integer. In this paper, we investigate a class of subfield codes of linear codes and obtain the weight distribution of \begin{equation*} \begin{split} \mathcal{C}_k=\left\{\left(\left( {\rm Tr}_1^m\left(ax^{p^k+1}+bx\right)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right) : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\}, \end{split} \end{equation*} where $k$ is a nonnegative integer. Our results generalize the results of the subfield codes of the conic codes in \cite{Hengar}. Among other results, we study the punctured code of $\mathcal{C}_k$, which is defined as $$\mathcal{\bar{C}}_k=\left\{\left( {\rm Tr}_1^m\left(a x^{{p^k}+1}+bx\right)+c\right)_{x \in \mathbb{F}_{p^m}} : \, a,b \in \mathbb{F}_{p^m}, \,\,c \in \mathbb{F}_p\right\}.$$ The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let $v_2(\cdot)$ denote the 2-adic order function and $v_2(0)=\infty$, the duals of $\mathcal{C}_k$ and $\mathcal{\bar{C}}_k$ are optimal with respect to the Sphere Packing bound if $p>3$, and the dual of $\mathcal{\bar{C}}_k$ is an optimal ternary linear code for the case $v_2(m)\leq v_2(k)$ if $p=3$ and $m>1$.