Collision detection & resolution: circle in a playfield of other circles and polygons - geometry

Question

I am working on a game that has a player sprite surrounded by a collision circle of a known radius. The player sprite can move about a playfield that consists of other sprites with their own collision circles and other obstacles made up of polygons. The other obstacles are rectangles at a 45 degree angle.
In addition, I want the player to adjust its movement when it does collide. I want the player to try to "push through" past the object instead of being stopped by it.
For example, if the player were to collide with another sprite's bounding circle, it would be stopped if its vector was exactly perpendicular to the tangent of the two circles' intersection.
However, if not perfectly perpendicular, the player would be, slowly at first, then faster, pushed along the tangent of the circle until it can continue past it unimpeded.
This works similarly when encountering one of the 45 degree rectangles.
What I need help with is the following: I am trying to find an analytic solution to detect both other sprites and obsticles, have the player's movement adjusted, and possibly stopped when adjusted to wedge between two or more objects.
I can do the collision detection and deflection for one object type at a time, but am struggling to put everything together into a comprehensive algorithm. I am currently working on an iterative pairwise resolution approach that "tries" different locations to result in a best-guess solution, but I really want a mathematically analytic solution. I'm hoping to have a function something like what appears in this psuedocode.
x = [player's x location]
y = [player's y location]
r = [player's collision radius]
// Array of other sprites on the playfield,
spr = [other sprites array]
// which contains 3 parameters, x, y, r. E.g., spr[3].x or spr[3].r,
// for the x position or collision radius for the fourth sprite in the
// array.
// Array of 45 degree rectangles on the playfield,
rect = [array of rectangles]
// which contain 4 parameters, x1, y1, x2, y2, the two opposite points
// of the rectangle. E.g., rect[0].x1, for the x position of the first
// point of the first rectangle.
// For simplicity, assume the above variables are all directly accessable
// in the function below.
// requestX and requestY is the position to which the player would
// like to move the player sprite.
definefunction collisionAdjustor(requestX, requestY) {
// Here I'd like to adjust the requested position if needed because
// of an intersection with one or more other sprites or rectangles.
// Finally return the location at which the player will actually be
// arriving.
return destinationX, destinationY
}
Any advice or suggestions would be much appreciated.
--Richard