Let $a,n$ be positive integers and let $p$ be a prime number. Let ${\mathbb{F}}_q$ be the finite field with $q=p^a$ elements. Let $\{a_i{\}}_{i=1}^{\mathrm{\infty }}$ be an arbitrary given infinite sequence of elements in ${\mathbb{F}}_q$ and $a_1\ne 0$. For each positive integer $i$, let $\{d_{i+j,i}{\}}_{j=0}^{\mathrm{\infty }}$ be an arbitrary given sequence of positive integers with $d_{ii}$ coprime to $q-1$. For each integer $n\ge 1$, let $N_n$, ${\overline{N}}_n$ and ${\tilde{N}}_n$ denote the number of ${\mathbb{F}}_q$-rational points of the hypersurfaces defined by the following three equations:

$$a_1{x}_1+\cdots +a_n{x}_n=b,$$

$$x_1^2+\cdots +x_n^2=b$$

and

$$a_1{x}_1^{d_{11}}+a_2{x}_1^{d_{21}}{x}_2^{d_{22}}+\cdots +a_n{x}_1^{d_{n1}}{x}_2^{d_{n2}}\cdots x_n^{d_{nn}}=b,$$

respectively. In this paper, we show that the generating function $\sum _{n=1}^{\mathrm{\infty }}{N}_n{t}^n$ is a rational function in $t$. Moreover, we show that if $p$ is an odd prime, then the generating functions $\sum _{n=1}^{\mathrm{\infty }}{\overline{N}}_n{t}^n$ and $\sum _{n=1}^{\mathrm{\infty }}{\tilde{N}}_n{t}^n$ are both rational functions in $t$. Moreover, we present the explicit rational expressions of $\sum _{n=1}^{\mathrm{\infty }}{N}_n{t}^n$, $\sum _{n=1}^{\mathrm{\infty }}{\overline{N}}_n{t}^n$ and $\sum _{n=1}^{\mathrm{\infty }}{\tilde{N}}_n{t}^n$, respectively.

Suppose ${\mathbb{F}}_q$ is a finite field having an odd characteristic. Let $D=\{x_1,\cdots ,x_{n-1},\mathrm{\infty }\}$, where $\{x_1,\cdots ,x_{n-1}\}⊊{\mathbb{F}}_q$. Assume that $k$ is an integer with $2\le k<n$. We show that $u(D)$ is a deep hole of Cauchy code $C(D,k)$ if $u(x)=\lambda (x-\delta {)}^{q-2}+\nu x^{k-1}+f_{\le k-2}(x)$, where $\lambda \in {\mathbb{F}}_q^{\ast }$, $\delta \in {\mathbb{F}}_q\setminus \{x_1,\cdots ,x_{n-1}\}$, $\nu \in {\mathbb{F}}_q$ and $f_{\le k-2}(x)\in {\mathbb{F}}_q[x]$ of a degree that does not exceed $k-2$. This expands the result shown in our previous paper. In particular, we also show that the received word $\mathit{u}=(u_1,\cdots ,u_n)\in {\mathbb{F}}_q^n$ is a deep hole of $C(D,k)$ if and only if the Lagrange interpolation polynomial of the first $n-1$ components of $\mathit{u}$ is $\lambda (x-\delta {)}^{q-2}+u_n{x}^{k-1}+u_{\le k-2}(x),$ where $\lambda \in {\mathbb{F}}_q^{\ast }$, $\delta \in {\mathbb{F}}_q\setminus \{x_1,\cdots ,x_{n-1}\}$, and $u_{\le k-2}(x)$ is a polynomial over ${\mathbb{F}}_q$ whose degree does not exceed $k-2$, if $\frac{q-1}{2}\le k<n\le q-2$.