Assume we have a smooth projective variety $X$ over a field (finite field if matters). Consider a smooth hypersurface section $H$ of $X$. Let $Spec(A)=U=X \setminus H$ be an affine variety. I want to know whether there are any nice description of vector bundles on $X$ such that they are trivial on $U$ and $H$? My suspicion was that they are all direct sums of line bundles of the form $O(nH)$ is that correct?

Edit: Let's assume that we are talking about rank $m$ vector bundles that are subsheaves of the constant sheaf given by the vector space $F(X)^{\oplus m}$ which $F(X)$ is the function field or equivalently field of fractions of $A$. Does that change anything?