This paper presents a complete solution to the measurement output feedback H-infinity control problem subject to asymptotic tracking of an infinite energy reference and the presence of nonproper output weights. Relying on generalised Sylvester-type equations, a set of all stabilising controllers is characterised, explicitly exhibiting the impacts of the reference signal and the output weight on the resulting controllers. Based on this specific parameterisation, additional disturbance attenuation is tackled, and both full-order and reduced-order controller synthesis are discussed. It is observed that the multiobjective problem is equivalent to a standard problem for an auxiliary system constructed from the original weighted system with the control input matrix and measurement output matrix being augmented by solutions to generalised Sylvester-type equations. An example is included to show the effectiveness of the present results.