Encoding a 4bit word (vector) $w$ can easily be done by $c := Gw$. Decoding a codeword of the Hamming(7,4) code is normally done by first computing $p = Hc$ where $p$ is zero, if there is no 1-bit error, or $p$ is the position of the 1-bit error.
Is it possible to have a matrix $K$ that automatically outputs the corrected word when multiplied by a codeword?
Endcoding and Decoding matrices in $\mathbb F_2$ $$G= \begin{pmatrix} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix} \qquad H = \begin{pmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \end{pmatrix}$$
A noted in the comments, you're looking for a matrix $K$ that outputs the corrected word when multiplied by a perturbed codeword, i.e. a word of distance $\le\!1$ from a codeword.
No such $K$ exists. For example, suppose you encode the word $w = 0000$ to get the codeword $c = Gw = 0000000$. The perturbed codewords are the words of weight $\le\!1$. But if $Kw = w = 0$ for all $x$ of weight $1$ then $K = 0$.