Let $X$ be a smooth projective curve over $\mathbb{C}$, $\infty,p\in X(\mathbb{C})$, and $L_n$ be the mixed Hodge structure of functions on $\pi_1(X-\{\infty\},p)$ that can be expressed by iterated integrals of length at most $n$. We will express the mixed Hodge extension arising from the weight filtration on quotient $L_n/L_{n-2}$ in terms of certain null-homologous algebraic cycles on $X^{2n-1}$. As a consequence, we will see that if $X,p,\infty$ are defined over a subfield $k\subset\mathbb{C}$, one can associate to this extension a family of
rational points on the Jacobian of $X$ parametrized by the Chow group $CH_{n-1}(X^{2n-2})$. When $n=2$, the results are due to Darmon, Rotger and Sols.