breaker size for 600' run using #4 wire
To answer the OP's question:

No you only need one breaker where the conductors recieve their supply. (At the main panel). The breaker needs to be sized the protect the conductors the well pump should have internal protection for the motor.

You didn't say whether your #4 was copper or aluminum. It makes a big difference. I will do both calculations with Ed Beal's numbers for the K factors (I usually use 10.4 for copper and 17.0 for aluminum which would yield smaller wires but warm wire has a higher K factor so I will use the hire numbers.)

A 2 hp 230 volt motor draws 12 amps according to the NEC. Using the voltage drop formula solved for area in circular mils. Acm= K x I x 2L /VD
With a 3% voltage drop the recommended number from the NEC yields 25,800 cm for copper which is a little smaller than #6 at 26,240 cm. so if you have #4 copper you have plenty of wire. 

However the same calculation with aluminum yields 42,400 and #4 is 41,740 so you would actually need #2 if it is aluminum. 

Voltage drops are recommendations from the National Electrical Code and are not an enforceable part of the code. Voltage drop calculations are performed with full load running currents not with startup currents. They recommend 3% for a branch circuit or feeder and 5% total between the two. Since this is a branch circuit the recommended amount is 3% or 7.2 volts. 

tl;dr You are good with #4 copper wire. The breaker size depends on the manufacturers recommendation for your well pump, or I need to know if you are using an inverse time breaker or instantaneous trip.