  Let $X$ be a smooth projective variety defined over a finite field. We show
that any algebraic $1$-cycle on $X$ is rationally equivalent to a smooth
$1$-cycle, which is a $\mathbb{Z}$-linear combination of smooth curves on $X$.
We also prove a generalized version of Poonen's Bertini theorem over finite
fields. Given a very ample line bundle $\mathcal{L}$ on $X$ and an arbitrary
line bundle $\mathcal{M}$, this version implies the existence of a global
section of $\mathcal{M}\otimes \mathcal{L}^{\otimes d}$ for sufficiently large
$d$ whose divisor is smooth.
