What action did the government take regarding the right of citizens to move the Court for restoring their Fundamental Rights during the Emergency?

It suspended this right.

Answer: It suspended this right.
Under the provisions of the Emergency, the right of citizens to move the Court to restore their Fundamental Rights was suspended.

The Emergency declaration on June 25, 1975, brought the agitation to an abrupt stop; strikes were banned; many opposition leaders were imprisoned; the political situation became very quiet though tense. Deciding to use its special powers under Emergency provisions, the government suspended the freedom of the Press. Newspapers were asked to get prior approval for all material to be published. This is known as press censorship. Apprehending social and communal disharmony, the government banned Rashtriya Swayamsevak Sangh (RSS) and Jamait-e-Islami. Protests and strikes and public agitations were also disallowed. Most importantly, under the provisions of Emergency, the various Fundamental Rights of citizens stood suspended, including the right of citizens to move the Court for restoring their Fundamental Rights.