The sums that you give are related to the number of points on

$y^2 = (x+a)(x+b)(x+c)$ and $y^2=(x+a)(x+b)(x+c)(x+d)$ respectively.

Assuming that $a,b,c,d$ are distinct, the first is an elliptic curve, and the second a curve of genus 2. By the "Riemann Hypothesis" for curves over finite fields each has the value

$a_p$ where $a_p$ is the trace of Frobenius. In the first case $|a_p| \le 2 \sqrt{p}$ and in the second $|a_p| \le 2 \lfloor 2 \sqrt{p} \rfloor$ (a result of Serre strengthening the normal RH).

[added: the formula I gave includes the "point at infinity" so that you need to subtract 1. The first case that you give corresponds to a curve of genus 0].

The sums that you give are related to the number of points on

$y^2 = (x+a)(x+b)(x+c)$ and $y^2=(x+a)(x+b)(x+c)(x+d)$ respectively.

Assuming that $a,b,c,d$ are distinct, the first is an elliptic curve, and the second a curve of genus 2 also a curve of genus 1. By the "Riemann Hypothesis" for curves over finite fields each has the value

$a_p$ where $a_p$ is the trace of Frobenius. In the both cases first case $|a_p| \le 2 \sqrt{p}$ and in the second $|a_p| \le 2 \lfloor 2 \sqrt{p} \rfloor$ (a result of Serre strengthening the normal RH) .

[added: the formula I gave includes the "point at infinity" so that you need to subtract 1 in the first case and 2 in the second. The first case that you give corresponds to a curve of genus 0].