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You can compute directly the intersection of the hyperplane and the polyhedron
sage: Hplane1 = Polyhedron(eqns=[(-3,2,2,2,-1,-1,-1,-1,-1,-1)])
sage: P = [[0, 0, 0, 0, 0, 0, 0, 0, 0], [-8/9, 1/9, 1/9, 1/9, ...
sage: P1 = Polyhedron(P)
sage: P1.intersection(Hplane1)
The empty polyhedron in QQ^9
So it looks that the hyperplane does not intersect your polyhedron.
