There are none.

Write it in the monomial basis with respect to $x_1,\ldots,x_k$, with coefficients being polynomials in $y_1,\ldots,y_n$. The only monomial basis functions with $k$ variables such that each variable occurs in at most $k-2$ terms are 1, $x_1^s+\cdots+x_k^s$ for $s\ge 1$, and $x_1^t x_2^t\cdots x_k^t$ for $t\ge 1$. (This is just because binomial coefficients are too big except near the ends.) The function $x_1^t x_2^t\cdots x_k^t$ has too high a degree, so the only chances have the form $$p(\mathbf{x},\mathbf{y}) = r(\mathbf{y}) + \sum_s q_s(\mathbf{y})(x_1^s+\cdots+x_k^s).$$ We must have $q_s(\mathbf{y})$ independent of $\mathbf{y}$ (i.e. constant) for each $s$, or else some $y_j$ occurs in at least $k$ monomials. But then $p$ is disconnected.

There are none.

Write it in the monomial basis with respect to $x_1,\ldots,x_k$, with coefficients being polynomials in $y_1,\ldots,y_n$. The only monomial basis functions with $k$ variables such that each variable occurs in at most $k-2$ terms are 1, $x_1^s+\cdots+x_k^s$ for $s\ge 1$, and $x_1^t x_2^t\cdots x_k^t$ for $t\ge 1$. (Because: if a monomial has $r$ variables there are at least $\binom kr$ similar monomials and each $x_j$ appears in at least $\frac kr \binom kr =\binom{k-1}{r-1}$ of them, which is more than $k-2$ unless $r\in\lbrace 0,1,k\rbrace$.) The function $x_1^t x_2^t\cdots x_k^t$ has too high a degree, so the only chances have the form $$p(\mathbf{x},\mathbf{y}) = r(\mathbf{y}) + \sum_s q_s(\mathbf{y})(x_1^s+\cdots+x_k^s).$$ We must have $q_s(\mathbf{y})$ independent of $\mathbf{y}$ (i.e. constant) for each $s$, or else some $y_j$ occurs in at least $k$ monomials. But then $p$ is disconnected.