  We show that the cubic nonconventional ergodic averages of any order with a
bounded multiplicative function weight converge almost surely to zero provided
that the multiplicative function satisfies a strong Daboussi-Delange condition.
We further obtain that the Ces\`{a}ro mean of the self-correlations and some
moving average of the self-correlations of such multiplicative functions
converge to zero. Our proof gives, for any $N \geq 2$,
$$\frac1{N}\sum_{m=1}^{N}\Big|\frac1{N}\sum_{n=1}^{N} \bnu(n) \bnu(n+m)\Big|
\leq \frac{C}{\log(N)^{\epsilon}},$$ and
$$\frac1{N^2}\sum_{n,p=1}^{N}\Big|\frac1{N}\sum_{m=1}^{N} \bnu(m)
\bnu(n+m)\bnu(m+p)\bnu(n+m+p)\Big| \leq \frac{C}{\log(N)^{\varepsilon}},$$
where $C,\varepsilon$ are some positive constants and $\bnu$ is a bounded
multiplicative function satisfying a Daboussi-Delange condition with
logarithmic speed. We further establish that the cubic nonconventional ergodic
averages of any order with Mangoldt weight converge almost surely provided that
all the systems are nilsystems.
