Regular global sections of invertible sheaves $\newcommand{\L}{\mathcal{L}}$
$\newcommand{\ox}{\mathcal{O}_X}$
Let $X$ be a projective scheme of dimension one over a field $k$ and let $\L$ be an invertible sheaf on $X$. 

What are sufficient conditions on $\L$ such that there exists $f \in \L(X)$ with $f\ox \cong \ox$.

What I have in mind: There is a bound only depending on $X$ such that $\L$ provides such a nice global section whenever $\deg_k \L := \chi(\L)-\chi(\ox)$ is greater than that bound.
Note that if $X$ is integral and we embed $\L$ into the constant sheaf $\mathcal{K}_X = K$ where $K$ denotes the function field of $X$, then obviously every non-zero global section of $\L$ satisfies the intended condition since it is no zero divisor. Moreover, then we may use Riemann-Roch to deduce the existence of a non-zero section for sufficiently large $\deg_k(\L)$.
 A: Assuming that my interpretation of the question is correct, and with the additional assumption that $X$ be Cohen-Macaulay, i.e., has no embedded points, then the answer is yes.
By Catanese, Franciosi, Hulek and Reid's Curve Embedding Theorem, it is enough that $\mathrm{deg}_B\mathcal L\geq 2p_a(B)+1$ holds for all generically Gorenstein sub-curves $B\subset X$. (This implies that $\mathcal L$ is very ample and very ample line bundles have regular sections.) See Catanese, Franciosi, Hulek, Reid, Embeddings of Curves and Surfaces, 1990, Nagoya Math. J. Vol. 154, p. 185-220.
It should be clear that there is no such bound depending only on $\mathrm{deg}_X(\mathcal L)$, though. $\mathcal{L}$ could be trivial on one component and ample on another component, giving positive degree but no section will be regular.
I have no idea what happens in the presence of embedded points.
Addendum: As pointed out by the OP, there could be a problem with finite fields. Here is a forceful way out. Let $Z\subset X$ be a $0$-dimensional closed sub-scheme meeting every component of $X$. If $X$ has no embedded points, then if a section is non-trivial at $Z$, then it is non-trivial at all generic points, hence regular. By the Curve embedding Theorem, if $\deg_B\mathcal L\geq 2p_a(B)-1+\mathrm{length}(Z\cap B)$  for all generically Gorenstein sub-curves $B\subset X$, then the restriction map $H^0(\mathcal L)\to H^0(L|_Z)\cong H^0(\mathcal O_Z)$ is surjective. Hence, the pre-image of a nowhere vanishing section is a regular section of $\mathcal L$.
