Must a ministry delete data when a citizen asks?

Short answer: No, but correction and security still apply

No. Section 17(4) says the erasure duty in Section 8(7) and the erasure right in Section 12(3) do not apply to processing by the State. Retention follows public records rules and schedules. Correction rights and all other duties, including security and breach reporting, still apply.

What the law says

Section 17(4) lifts DPDP erasure for the State. Public records rules govern destruction.

Section 8(7) · Rule 8: Erase personal data when its purpose is over or consent is withdrawn, unless a law requires you to keep it, and have your processors erase it too. Rule 8(3) asks every Data Fiduciary to keep personal data, traffic data and logs for at least one year for purposes listed in the Rules.
Sections 11–14 · Rule 14: People can ask for a summary of their data and who it was shared with, ask for correction, completion, updating or erasure, complain, and nominate someone to act for them. You must publish how to do this.
Section 8(3): If personal data is used to make a decision about a person, or is shared with another Data Fiduciary, reasonable care must be taken that it is complete, accurate and consistent.

Steps

Apply record retention schedules.
Correct errors when asked.
Tell the citizen why data is kept.
Destroy records as the schedule allows.
Keep access limited while records are kept.

Evidence to keep

Retention schedule
Correction log
Destruction register

Common mistakes

Thinking no duties apply
Ignoring correction requests
Keeping open access to old records

From each seat

Data protection nodal officer: Use this in replies to erasure requests.
Legal cell: Confirm the schedules that apply.
Customer service department: Explain politely why data stays.

What a good answer from management sounds like

“We follow retention schedules, correct errors on request and explain retention in replies.” Effort and time: Light.