Prove that the diophantine equation $(xz+1)(yz+1)=az^{k}+1$ has infinitely many solutions in positive integers.

This answer is based on the excellent work of Will Jagy. This solves all cases of $k>3.$

Let $p<k$ be an odd prime such that $p\not\mid k.$

Solve $kd\equiv -1\pmod{p}.$ Let $n=(kd+1)/p.$ Note that since $p<k,$ $n>d.$

Then for any integer $t,$ we can take $z=a^{d}t^p$ so that $$\begin{align}az^k+1&=a^{kd+1}t^{kp}+1\\&=\left(a^nt^k\right)^p+1\\ &=(a^nt^k+1)\left(1+a^nt^k\sum_{j=1}^{p-1} (-1)^j\left(a^nt^k\right)^{j-1}\right) \end{align}$$

Where the last equation is because when $p$ is odd, $$ \begin{align}u^p+1&=(u+1) \sum_{j=0}^{p-1} (-1)^ju^j \\&=(u+1)\left(1+u\sum_{j=1}^{p-1}(-1)^ju^{j-1}\right)\end{align}$$

Now, since $n>d,$ we can set $$ \begin{align}x&=a^{n-d}t^{k-p}\\ y&=a^{n-d}t^{k-p}\sum_{j=1}^{p-1} (-1)^j\left(a^nt^k\right)^{j-1} \end{align}$$

For $k\geq 4$ we can always find such a $p$ by taking a prime factor of $n-1$ or $n-2$ if $n$ is even or odd, respectively.

So this solves all cases $k>3.$

You don't need $p$ prime, just that $1<p<k$ is odd and $\gcd(p,k)=1.$

k even

So when $k$ is even, we can take $p=k-1.$ Then $d=p-1$ and $n=p.$

Then for any integer $t,$ $$\begin{align}z&=a^{k-2}t^{k-1}\\x&=at\\y&=at\sum_{j=1}^{k-2}(-1)^j\left(a^{k-1}t^k\right)^{j-1}.\end{align}$$

k odd

Likewise, if $k=2m+1$ is odd, then you can take $p=2m-1,$ $d=m-1$ and $n=m.$ Then for any integer $t$:

$$\begin{align}z&=a^{m-1}t^{2m-1}\\ x&=at^2\\ y&=at^2\sum_{j=1}^{2m-2}(-1)^j\left(a^mt^{2m+1}\right)^{j-1} \end{align}$$

is a solution.

In particular, for $k>3$ there are infinitely many solutions $(x,y,z)$ with $a\mid x$ and $x\mid y$ and $x\mid z.$

Getting there. Here is $k=4.$ a family of solutions to $$ a z^4 + 1 = (xz+1)(yz+1) $$

is parametrized by integer $t$ with

$$ y=at $$

$$ x = a^4 t^5 - at $$

$$ z = a^2 t^3 $$

Both sides of the equation are $$ a^9 t^{12} + 1 $$

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For that matter, we can take care of all $k \neq 0 \pmod 3$ this way.

When $k > 3$ and $k \equiv 1 \pmod 3,$ we may take $$ y = a^{\frac{2k-5}{3}} \; t^{k-3} $$ $$ z = a^2 t^3 $$ followed by $$ x = y \left( y^2 z^2 - 1 \right) $$

When $k > 3$ and $k \equiv 2 \pmod 3,$ we may take $$ y = a^{\frac{k+1}{3}} \; t^{k-3} $$ $$ z = a t^3 $$ followed by $$ x = y \left( y^2 z^2 - 1 \right) $$