Can I use background with texture?

This is okay.

Animate[Plot3D[h x^2 y, {x, -1, 1}, {y, -1, 1}, PlotRange -> 2, 
  Background -> Blue], {h, 0, 1, .01}]

But I want a background, in the animation, same the picture in the file attachment. Is it possible?

You can experiment with Overlay.

img = Import["~/Downloads/Background.png"];

Animate[Overlay[{
   img,
   Plot3D[h x^2 y, {x, -1, 1}, {y, -1, 1},
    PlotRange -> 2,
    Axes -> None,
    ImageSize -> ImageDimensions@img/2]}],
 {h, 0, 1, .01}]

Dear Rohit
Thank you very much.