When we are trying to find out constants for Newton's interpolating polynomial, we use divided difference method to find the constants. Then we have Hermite-Genocchi formula to find those constants when one or more nodes coincide.
What is the meaning of nodes coinciding, does that mean we have same (x,y) more than 1 times recorded in our observation or different 'y(x)' for the same value of x.
It would have been nice if you had written down your version of the formulas. The most general I could find is \begin{multline} [u_0,u_1,…,u_n]q\\=\int_{0\le x_n\le x_{n-1}\le…\le x_1\le 1}q^{(n)}\Bigl(u_0+(u_1-u_0)x_1+(u_2-u_1)x_2+\\…+(u_n-u_{n-1})x_n\Bigr)\,d(x_1,x_2,…,x_n) \end{multline} This has, at this point, nothing to do with the interpolation of empirical data. However, it might give a nicer picture to approximate a cluster of data points with a linear or quadratic regression and then use these values and derivatives in a Hermite-interpolation. You would have to compare such a procedure to performing polynomial regression with the full data set in one step.
As to repeating points: This means that you will need derivative values in the interpolation formula. In the most simple case you get for a close, but not identical $$ [x-δ,x+δ]q=\frac{q(x+δ)-q(x-δ)}{2δ} $$ so that in the limit $[x,x]q=q'(x)$ and for instance $$ [x,x,y]q=\frac{[x,y]q-[x,x]q}{x-y}=\frac{q(x)-q(y)-q'(x)(x-y)}{(x-y)^2} $$