What is the impedance of this unusual (magnetically coupled) circuit?
For a perfect transformer, the primary X1 would appear to be shorted because the secondary X2 is shorted.  The impedance would therefore simply be Z1 + R1.

X1 appears in the circuit only to the extent that the transformer is not ideal.  There are 4 main contributions to the impedance of X1:

The DC resistance of X1.  This appears like a resistance in series with ideal X1.

The DC resistance of X2.  This appears like a resistance in series with ideal X2.  What that means reflected back to X1 depends on the turns ratio.  Whatever it is, is can be thought of as that reflected resistance in series with X1.

The leakage inductance of X1.  This is the magnetic field created by X1 that is not coupled to X2.  It appears like a inductor in series with X1.

The leakage inductance of X2, which can be thought of as a uncoupled inductor in series with X2.  Like the DC resistance of X2, this is reflected back to X1 according to the turns ratio.