Why the extra transistors in this current mirror?
Q3 and Q4 are not "Diode connected".

They are arranged as emitter followers to reduce the error that would result from the base current for Q1 and Q2 being derived from the current reference flowing through R2.

The gain of Q3,Q4 will reduce the current required by a factor of Hfe**2 (maybe 10,000 times) so that virtually all the current from R2 flows through Q1 rather than some being diverted.

They will also mean that Q1 is operating at a higher Vce (two Vbe drops) this will affect the current through R2 but may put Q1 into a better operating point.