We study the local risk-minimization approach for defaultable claims with random recovery at default time, seen as payment streams on the random interval aEuroe0,tau a TaEuro >, where T denotes the fixed time-horizon. We find the pseudo-locally risk-minimizing strategy in the case when the agent information takes into account the possibility of a default event (local risk-minimization with G-strategies) and we provide an application in the case of a corporate bond. We also discuss the problem of finding a pseudo-locally risk-minimizing strategy if we suppose the agent obtains her information only by observing the non-defaultable assets.