High-voltage direct-current (HVDC) grids may provide fast frequency support to ac grids with the aid of supplementary control algorithms and synthetic inertia contribution from offshore wind farms. However, when all converters within the HVDC grid are fitted with these supplementary controllers, undesirable power flows and reduced power transfers may occur during a power imbalance. This is due to simultaneous frequency oscillations on the different ac systems connected to the HVDC grid arising during the support operation. To prevent this adverse effect, an auxiliary dead-band controller (ADC) is proposed in this study. The ADC modifies the dead-band set-point of the fast frequency controllers using measurements of the rate of change of frequency and frequency deviation. A four-terminal HVDC integrated with an offshore wind farm is modelled to analyse and study the effectiveness of three different supplementary fast frequency control algorithms. Results show that the proposed ADC scheme improves the performance of fast frequency control algorithms. For completeness, a small-signal stability analysis is carried out to confirm that a stable system operation is maintained.