= Solution
In <fixed-design nonparametric regression>, the covariates $x_i$ are deterministic and
$$
Y_i=m(x_i)+\varepsilon_i,\qquad \mathbb E\varepsilon_i=0.
$$
For the usual finite-variance model assume independent errors with $\operatorname{Var}(\varepsilon_i)\le\sigma^2$. In <random-design nonparametric regression>, the pairs $(X_i,Y_i)$ are independent observations and $m(x)=\mathbb E[Y_i\mid X_i=x]$; equivalently $Y_i=m(X_i)+\varepsilon_i$ with conditional mean-zero errors. Conditional <variances> are generally assumed bounded for risk bounds.
For a nonnegative <regression kernel> $K$ and a <smoothing bandwidth> $h>0$, the <Nadaraya–Watson estimator> is
$$
\widehat m_{\mathrm{NW}}(x)=\frac{\sum_i K((x_i-x)/h)Y_i}{\sum_iK((x_i-x)/h)},
$$
when the denominator is positive; replace $x_i$ by $X_i$ for <random-design nonparametric regression>. The <local polynomial regression> <estimator> of degree $\ell$ minimizes the <weighted least squares> criterion
$$
\sum_i K(t_i)\left(Y_i-\sum_{j=0}^\ell\beta_j\frac{t_i^j}{j!}\right)^2,
\qquad t_i=\frac{x_i-x}{h},
$$
and reports $\widehat\beta_0$. If the weighted <Gram matrix> is invertible, this minimizer is unique. \b[The Nadaraya-Watson <estimator> is exactly <local polynomial regression> of degree zero.] Higher degrees reproduce nonconstant <polynomials> and reduce boundary <bias of an estimator>.
The requested error bound needs \b[$\ell\ge s-1$ and mean-zero independent errors of uniformly bounded <variance>], along with the matrix condition in the hint. The degree requirement is missing from the printed question. To see the problem, take $\ell=0$, $K=\mathbf1_{[-1,1]}$, $m(t)=t$, zero errors, $x=0$, and $h=n^{-1/4}$. This <regression function> has two bounded <derivatives>, but the local constant estimate is the average of $i/n$ for $1\le i\le N=\lfloor nh\rfloor$:
$$
\widehat m(0)-m(0)=\frac{N+1}{2n}\sim\frac h2.
$$
The scalar matrix $B=N/(nh)$ is bounded away from zero for every $n$. Nevertheless $(nh)^{-1/2}+h^2=n^{-3/8}+n^{-1/2}=o(h)$, so the printed conclusion with $s=2$ fails. A bounded-variance error hypothesis is also essential; otherwise an <expectation> bound need not even be finite.
Under the qualified hypotheses, use $U(t)=(1,t,\ldots,t^\ell/\ell!)^T$ and
$$
B_n=\frac1{nh}\sum_iU(t_i)U(t_i)^TK(t_i),\qquad
W_{ni}(x)=\frac1{nh}U(0)^TB_n^{-1}U(t_i)K(t_i).
$$
This symmetric <Gram matrix> has inverse <operator norm> bounded by the reciprocal of its uniform positive eigenvalue lower bound. If $K$ is supported in $[-R,R]$, only $|x_i-x|\le Rh$ contribute. The equally spaced design and $nh\ge1$ imply that there are at most $2Rnh+1\le Cnh$ such points. The bounded kernel and bounded $U$ on this interval therefore yield
$$
\max_i|W_{ni}(x)|\le\frac C{nh},\qquad
\sum_i|W_{ni}(x)|\le C,\qquad
\sum_i W_{ni}(x)^2\le\frac C{nh}.
$$
For the stochastic part, <variance additivity for independent random variables> and <Cauchy-Schwarz inequality> give
$$
\mathbb E\left|\sum_iW_{ni}(x)\varepsilon_i\right|
\le\left(\sum_iW_{ni}(x)^2\operatorname{Var}\varepsilon_i\right)^{1/2}
\le C\sigma(nh)^{-1/2}.
$$
For the deterministic part, let $Q_x$ be the <Taylor polynomial> of $m$ at $x$ of degree $s-1$. The <polynomial reproduction property of local polynomial regression> gives $\sum_iW_{ni}(x)Q_x(x_i)=Q_x(x)=m(x)$. By <Taylor's theorem>, $|m(x_i)-Q_x(x_i)|\le C|x_i-x|^s\le Ch^s$ on the kernel window. Thus
$$
\left|\sum_iW_{ni}(x)m(x_i)-m(x)\right|
\le Ch^s\sum_i|W_{ni}(x)|\le Ch^s.
$$
Combining the two parts proves
$$
\boxed{\mathbb E|\widehat m_n(x)-m(x)|\le C\bigl((nh)^{-1/2}+h^s\bigr),\qquad \ell\ge s-1.}
$$
The argument also applies at the endpoints when the assumed matrix bound holds. With arbitrary degree, the generally valid smoothness term is $h^{\min(s,\ell+1)}$; extra symmetry can improve some interior biases but does not fix the general boundary counterexample.
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