Why is a strake placed on the fuselage in front of the tailplane on the Hawk aircraft?
You may have noted that the tailplane of the Hawk is a full-flying surface, also called an all-moving surface. When the stick is pulled aft, the leading edge will align with the strake, forming in effect a leading edge extension. This will produce a vortex over the lower surface of the inner tailplane and prevent early flow separation at the tailplane root at low speed.

The obvious next question is: Why is it not part of the tailplane itself? Adding it to the tailplane would increase drag and produce nonlinear control forces. Since it was added later in the life of the design, fixing it to the fuselage allowed to keep the tail surfaces and their actuation unchanged. 

Note that early Hawk versions (Mk.1, Hawk 50) lacked this detail. In flight testing, it was found that the Hawk would suddenly nose-dive with full flaps and gear up. This was dubbed "phantom dive". Cutting back the flaps solved that behavior, but later, heavier versions of the Hawk needed more lift, so a different fix had to be found. From the Hawk story by Harry Fraser-Mitchell:


  It was shown with the half model of the Hawk at Hatfield that high
  local downwash at the tail, coupled with the very large nose down
  pitching moment induced by the flap, was causing the tailplane to
  stall on its lower surface, so that it could no longer provide
  adequate balancing power. It needed more lift, extended to higher
  angles of attack.
  
  In the case of the F-4K Phantom aircraft this was achieved by
  installing a fixed leading edge slot to the tailplane, harking back to
  demonstrations of such devices by Handley Page on wings in the early
  Twenties! A fixed slot with its associated drag was not an option on
  the Hawk although a cambered tailplane was tried on the model with
  some success. Removal of the outboard vane of the flap reduced the
  flap pitching moment to such a value that the standard tailplane could
  cope, so this was the quick solution for the RAF. However, for the US
  Navy VTX project, (and for later combat versions of the Hawk) the
  maximum possible lift was required, so that at least the outer flap
  vane had to be replaced. The dive phenomenon had to be fixed.
  
  An example of the cross-fertilization of knowledge due to the matrix
  working of the design department now occurred. Barry Pegram, then
  Section leader of the Fluid Dynamics section of the Aerodynamics
  Department, had been working on the adoption of leading edge root
  extensions (LERX) for the “Harrier” wing, and this work had shown that
  these devices extended the lift of the wing to higher angles of attack
  by virtue of the non-linear lift developed by the vortex flow they
  created. He proposed that these should be added to the tailplane of
  the Hawk model, but this would have had a serious effect on the
  tailplane hinge moments. The author, who was working with Barry in the
  V/STOL tunnel at Hatfield, suggested that the ‘tailplane canard vane’
  (TCV), as it was called, could be fixed to the fuselage at such a
  position that it was lined up with the flow at normal conditions, but
  with its trailing edge adjacent to, with a small clearance, the
  leading edge of the tailplane at its maximum nose down position. 
  
  Experimenting showed that these vanes could be made quite small and
  they gave a complete cure to the problem in the wind tunnel, with very
  little drag in the normal flight regime. To prove the concept in
  flight, some temporary vanes were manufactured which could rapidly be
  fitted to one of the test aircraft which had flaps with the full vane.
  First, the aircraft was flown without the TCV to establish the
  conditions under which the ‘Phantom Dive’ occurred on that particular
  aircraft. On a later flight, the TCVs were fitted and the aircraft
  flown again to the identical conditions as before. Despite every
  attempt by the pilot to instigate the phenomenon, it did not occur –
  the vanes were a complete success, even though they looked
  inconspicuously small for such a large effect.