Can a moving current carrying wire produce electric field? As we know, a current carrying wire can create a magnetic field which is perpendicular to the direction of current (From relativity, we can understand why magnetic fields are produced ) and the net charge of the wire is zero. So, no electric field will be produced. 

If the wire is moving, will the magnetic field move?  I learned that a moving magnetic field can produce electric field. Where am I missing the concept clearly?  
 A: 
the net charge of the wire is zero. So, no electric field will be produced.

The charge of the wire being zero means that there is no electrostatic E-field. However there are other ways to produce E fields. Faraday’s law says $$\nabla \times E=-\frac{\partial}{\partial t}B$$
Which means that you can also get a circulating E field by having a B field which changes in time. 
In your case, since the wire is moving the B field changes over time leading to an E field. 
A: To explore the concept further, note that the overall charge density of the wire is in fact something that depends on your frame of reference. 
To see this, imagine that you're on a starship next to an infinitely long, neutrally-charged but current-carrying wire. As you accelerate closer to the speed of light, you "catch up" to the electrons moving in the wire -- but conversely, the 'stationary' positive ions they're loosely attached to are now moving backwards. 
Since they're moving relative to you, they now experience length contraction; meanwhile, the electrons, which were already moving and therefore already contracted relative to your original inertia frame, are spreading back out as you catch up. So the relative charge densities will themselves change as you move -- meaning that the wire appears to pick up a charge!
And, of course, whether you're moving relative to the wire or it's moving relative to you doesn't matter -- so you immediately see that yes, a moving neutrally-charged wire should general produce an electric field.
(This is somewhat separate to the Ampere's Law effect from Dale, though they turn out to be the same law when you move into the full spacetime formalism.)
