Maybe this? But you may be more specific about what you need.

\documentclass{article}
\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}
\node[and port, number inputs=1] (A) at (0, 0) {};
\node at (A) [ocirc,fill=black] {};
\node[left] at (A.in 1) {(x_1)};
\node[left] at (A.in 2) {(x_2)};
\node[or port] (B) at (4,0.28) {};
\node[left] at (B.in 1) {\(x_3\)};
\node[right] at (B.out) {\((x_1'x_2)'+x_3\)};
\draw (A.out) -- (B.in 2);
\end{circuitikz}
\end{document}
EDIT
So maybe this one?

\documentclass{article}
\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}
\node[and port, number inputs=1] (A) at (0, 0) {};
\node at (A) [ocirc,fill=black] {};
\node[left] at (A.in 1) {(x_1)};
\node at (A.left) [yshift=8pt,ocirc,fill=black] {};
\node[left] at (A.in 2) {(x_2)};
\node[or port] (B) at (3,-.65) {};
\node[left] at (B.in 2-| A.in 1) (x3) {\(x_3\)};
\node[right] at (B.out) {\((x_1'x_2)'+x_3\)};
\draw (A.out) --++ (1,0) |- (B.in 1);
\draw (x3) -- (B.in 2);
\end{circuitikz}
\end{document}
SECOND EDIT
Following OP's demand, I propose another evolution of the solution, but it's now mandatory to scale the OR gate a bit.

\documentclass{article}
\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}
\node[and port, number inputs=1] (A) at (0, 0) {};
\node at (A) [ocirc,fill=black] {};
\node[left] at (A.in 1) {(x_1)};
\node at (A.in 1) [xshift=6.5pt,ocirc,fill=black] {};
\node at (A.left) [yshift=8pt,ocirc,fill=black] {};
\node[left] at (A.in 2) {(x_2)};
\node[or port,scale=1.2] (B) at (3,-.35) {};
\node[left] at (B.in 2-| A.in 1) (x3) {\(x_3\)};
\node[right] at (B.out) {\((x_1'x_2)'+x_3\)};
\draw (A.out) -- (B.in 1);
\draw (x3) -- (B.in 2);
\end{circuitikz}
\end{document}
\node at (A) [ocirc,fill=black] {};. For the rest of the question, I didn't understand exactly what you want. Maybe if you drew a sketch of it? – SebGlav Apr 29 '21 at 21:47\node[and port, yscale=0.75] (A) at (0, 0) {};. This scales the y dimensions of the gate by 0.75 times which is enough that there is no overlap and the circuit is still mostly the same. – Willoughby Apr 30 '21 at 21:54x_3input – Manjoy Das Apr 30 '21 at 22:03