For the line bundle $L$ over a manifold $M$, the connection $\nabla \colon L \to \Omega_M^1 \otimes L$ is defined. To the best of my knowledge when $L$ is a trivial line bundle ${\cal O}_M$, $\nabla$ descends to the usual differential $d \colon {\cal O}_M \to \Omega_M^1$. I cannot make head or trail of what $\nabla$ is all about for a general line bundle $L$.
In order to fix the idea, let us choose a very ample line bundle $L \colon= {\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$ of the projective line ${\Bbb P}_{\Bbb C}^1$ whose total space is ${{\Bbb P}_{\Bbb C}}^2 \setminus {(0,0,1)}$. The global section of ${\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$ is the circle on ${\Bbb P}_{\Bbb C}^2$ which does not pass through $(0,0,1)$. c.f. Global section of the line bundle ${\cal O}_{{\Bbb P}_k}(1)$ over ${\Bbb P}_k^1$.
Q. What is the Levi-Civita connection $\nabla$ for $L = {\cal O}_{{\Bbb P}_{{\Bbb C}}^1}(1)$? Is there any intuitive figure of $\nabla \colon {\cal O}_{{\Bbb P}_{\Bbb C}^1}(1) \to \Omega_{{\Bbb P}_{\Bbb C}^1} \otimes_{{\cal O}_{{\Bbb P}_{\Bbb C}}^1} \!\!\!{\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$ in view of the global section $s$, i.e., circle on ${\Bbb P}_{{\Bbb C}}^2$, of ${\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$? What if we choose other connection if it should exist?
Question: What is the Levi-Civita connection $\nabla$ for $L = {\cal O}_{{\Bbb P}_{{\Bbb C}}^1}(1)$? Is there any intuitive figure of $\nabla \colon {\cal O}_{{\Bbb P}_{\Bbb C}^1}(1) \to \Omega_{{\Bbb P}_{\Bbb C}^1} \otimes_{{\cal O}_{{\Bbb P}_{\Bbb C}}^1} \!\!\!{\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$ in view of the global section $s$, i.e., circle on ${\Bbb P}_{{\Bbb C}}^2$, of ${\cal O}_{{\Bbb P}_{\Bbb C}^1}(1)$? What if we choose other connection if it should exist?
Answer: The Levi-Civita connection is defined for the tangent bundle of a Riemannian manifold - it is by definition the unique connection compatible with the Riemannian metric.
Your eample above asks for an example of a connection a line bundle $L(d):=\mathcal{O}(d)$ with $d=1$ on the complex projective line $C:=\mathbb{P}^1_{\mathbb{C}}$. Let $k:=\mathbb{C}$. There is the well known Atiyah sequence
$$(*)\text{ } 0 \rightarrow \Omega^1_{C/k}\otimes L(d) \rightarrow J^1(L(d)) \rightarrow L(d) \rightarrow 0$$
which is an exact sequence of left $\mathcal{O}_C$-modules. It is split by a holomorphic connection
$$\nabla: L(d) \rightarrow \Omega^1_{C/k} \otimes L(d).$$
In the case when $d=1$ it follows the sequnce does not split, hence $L(1)$ does not have a holomorphic connection. For the projective line it follows $\Omega^1_{C/k} \cong L(-2)$ and you may check this claim by explicit calculations. If a line bundle $L$ on a curve $C$ has a connection it follows the set of connections on $L$ is the set
$$ Hom_{\mathcal{O}_C}(L, \Omega^1_{C/k}\otimes L) \cong Hom_{\mathcal{O}_C}(\mathcal{O}_C, \Omega^1_{C/k}) \cong H^0(C, \Omega^1_{C/k})$$
and in the case of the projective line this is equals $(0)$ since $\Omega^1_{C/k} \cong L(-2)$.
Note: You should specify: Do you view the complex projective line $C$ as a complex manifold or as a real smooth manifold (or scheme over $\mathbb{R}$)? The Weil restriction of the complex projective line is isomorphic to the real 2-sphere:
$$Res_{\mathbb{C}/\mathbb{R}}(\mathbb{P}^1_{\mathbb{C}}) \cong S(2),$$
and the real smooth vector bundle $E$ associated to $L(1)$ is a real rank 2 smooth vector bundle on $S(2)$, and this always has a smooth connection.
https://en.wikipedia.org/wiki/Levi-Civita_connection