Let X ⊂ PN be a projective, non-degenerate, irreducible smooth variety of dimension n. After giving the definition of generalised OADP-variety (one apparent double point), i.e. varieties X such that: ◦ n(k + 1) − (N − r)(k − r) + r = N, ◦ there is one apparent (k +1)-secant (r−1)-space to a generic projection of X from a point, we concentrate in studying generalised OADP-surfaces in low dimensional projective spaces, and the main result of this paper is the classification of smooth surfaces in P6 with one 4-secant plane through the general point of P6.