We study the Brauer-Schur-Weyl duality between the quotient
$\bb_n(-2m)/\bb_n^{(f)}$ of the Brauer algebra $\bb_n(-2m)$ and the symplectic
group $Sp(V)$ on the space $\mathcal{HT}_n^{\otimes f}$ of partially harmonic
tensors of valence $f$ in $V^{\otimes n}$, where $\bb_n^{(f)}$ is the two-sided
ideal generated by $e_1e_3... e_{2f-1}$ and $1\leq f\leq [n/2]$. We prove that
the dimensions of $\mathcal{HT}_n^{\otimes f}$ and the endomorphism algebra
$\End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\bb_n^{(f)}\Bigr)$ are both
independent of $K$, and the natural homomorphism from $\bb_n/\bb_n^{(f)}$ to
this endomorphism algebra is always surjective. We show that the left
$Sp(V)$-module $\mathcal{HT}_n^{\otimes f}$ has a Weyl filtration and is
isomorphic to the dual of $V^{\otimes n}\bb_n^{(f-1)}/V^{\otimes n}\bb_n^{(f)}$
as a $Sp(V)$-$(\bb_n/\bb_n^{(f)})$-bimodule. We also obtain a
$Sp(V)$-$\bb_n$-bimodules filtration of $V^{\otimes n}$ such that each
successive quotient is isomorphic to some $\nabla(\lam)\otimes z_{g,\lam}\bb_n$
with $\lam\vdash n-2g$, $\ell(\lam)\leq m$ and $0\leq g\leq [n/2]$, where
$\nabla(\lam)$ is the co-Weyl module associated to $\lam$ and $z_{g,\lam}$ is
an explicitly constructed maximal vector of weight $\lam$ in $V^{\otimes n}$.