The Hasse principle basically addresses the question of whether an algebraic variety defined over a global field $K$ has any $K$-rational points at all. For cubic equations(curves and varities), this was a topic of much research. The most commonly studied obstruction to Hasse principle is the Manin obstruction, or the Brauer-Manin group. See for example the last chapter of Y. I. Manin's book "Cubic forms : algebra, geometry and arithmetic", preview available for free on google books. Others too have taken up this line of thought and come up with various other results -- for instance, in some cases the Manin obstruction is insufficient to explain the nonexistence of rational points. This is an active topic of research. Refer to the works of David Harari, Bjorn Poonen, Alexei Skorobogatov, among others, for examples of results obtained in the last few decades.
It appears to me that you have a misunderstanding about definitions. The Tate-Shafarevich group is an object attached to an elliptic cuve $E$ over a number field $K$. It can be equivalently described as equivalence classes of all torsors of $E$ that have $K_v$-rational points for all places $v$ for $K$, but no $K$-rational point, modulo the equivalence relation of equivalence as $E$-torsors, the isomorphisms being defined over $K$. Here it so happens that the given elliptic curve is the Jacobian variety of the torsor, the latter being a genus $1$ curve over $K$ possibly without a $K$-rational point. The definition of an elliptic curve $E/K$ is that it is a genus $1$ curve $E$ defined over $K$ with a specified rational point(the origin), and therefore you do not need to consider the Hasse principle for the elliptic curve itself; it is a moot question; and your statement of "fixing" the Hasse principle for elliptic curves is meaningless.
Lastly, the phrase "error term" is used in analytic expression involving some asymptotic results or approximations, such as in the prime number theorem. In the algebraic context that phrase is strange. You should rather say "obstruction", "obstruction group", etc..