Does a field have any physical meaning or significance? Is the concept of a field just a mathematical construct? Is there any way to realize its existence? For instance, the fact that moving a charge affects other charges in the surrounding not instantaneously is explained in terms of the existence of an electric field. Is there no other approach to this problem?
 A: 
Does a field have any physical meaning or significance?

Yes. See Einstein talking about field theory in 1929 and note this: "The two types of field are causally linked in this theory, but still not fused to an identity. It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric". This is Einstein saying a field is a state of space. When you play around with a couple of magnets feeling attraction and repulsion, it's because of the space between those magnets has a different state to the space between my two hands. It's similar for a gravitational field. The state of space in the room you're in is different to the state of space way away from any stars or planets.   

Is the concept of a field just a mathematical construct? 

No. See above.   

Is there any way to realize its existence?

Yes, you let go of your electron, or your pencil. If it stays put you tend to say there's no field there. 

For instance, the fact that moving a charge affects other charges in the surrounding not instantaneously is explained in terms of the existence of an electric field. Is there no other approach to this problem?

Yes and no. See the Wikipedia Coulomb Law article and note this: "An electric field is a vector field that associates to each point in space the Coulomb force experienced by a test charge". Also see the Wikipedia electromagnetic field article: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field". The electric field describes the force between two (or more) charged particles. It doesn't describe the state of space where one particle is. The electromagnetic field does that, and the force is the result of the interaction of two electromagnetic fields. Remember this: a single charged particles doesn't feel any force at all, because it takes two to tango.  
A: Field is a mathematical notion. It is possible to formulate electromagnetic theory with particles in such a way that the field is not needed. This is due to Fokker and Tetrode, I think. More accessible reference is the paper (the part about the absorber is additional, not needed for the theory to work without fields):
J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct
Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.
http://dx.doi.org/10.1103/RevModPhys.21.425
