= Solution
At the Gaussian fixed point, the <engineering dimension> of the field is
$$
[\phi]=\frac{d-2}{2}.
$$
The operator $\phi^n(\nabla^2\phi)^m$ contains $n+m$ fields and $2m$ derivatives. Since its integral must be dimensionless,
$$
[g_{n,m}]
=d-\frac{(n+m)(d-2)}2-2m.
$$
It is marginal when this vanishes, namely
$$
d=\frac{2(n-m)}{n+m-2},
$$
when $n+m\ne2$. Since $n,m$ are positive, the exceptional case is $n=m=1$: $\phi\nabla^2\phi$ has a dimensionless coupling in every $d$ and differs from the kinetic term by integration by parts. If the displayed formula gives no positive $d$, there is no positive spatial dimension in which that operator is naively marginal.
At an interacting fixed point, field and composite operators acquire <anomalous dimensions>, and operators with the same symmetries can mix under renormalization. Their full <scaling dimensions> can therefore differ from these naive engineering dimensions.
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