Exhibit 99.4
Step 1: Step 2: Step 3: Step 4: Step 5: Go to
www.envisionreports.com/ghrs. Click on Cast Your Vote or Request Materials. Follow the instructions on the screen to log in. Make your selections as instructed on each screen for your delivery preferences. Vote your
shares. www.envisionreports.com/ghrs Easy Online Access — View your proxy materials and vote. Online Go to www.envisionreports.com/ghrs or scan the QR code — login details are located in the shaded bar below. Annual General Meeting
Notice 1234 5678 9012 345 Important Notice Regarding the Availability of Proxy Materials for the GH Research PLC Annual General Meeting to be Held at 2:00 p.m. Irish Standard Time on July 31, 2025 at Joshua Dawson House, Dawson Street,
Dublin 2, D02 RY95, Ireland Under Securities and Exchange Commission rules, you are receiving this notice that the proxy materials for the annual shareholders’ meeting are available on the Internet. Follow the instructions below to view the
materials and vote online or request a copy. The items to be voted on and location of the annual meeting are on the reverse side. Your vote is important! This communication presents only an overview of the more complete proxy materials that
are available to you on the Internet. We encourage you to access and review all of the important information contained in the proxy materials before voting. The Notice of Annual General Meeting, Irish Statutory Financial Statements and the
Form 20-F are available at: 045XYB When you go online, you can also help the environment by consenting to receive electronic delivery of future materials. Obtaining a Copy of the Proxy Materials – If you want to receive a copy of the proxy
materials, you must request one. There is no charge to you for requesting a copy. Please make your request as instructed on the reverse side on or before 21 July 2025 to facilitate timely delivery. 2 N O T The Sample Company Votes
submitted electronically must be received by 11:59 P.M., Irish Standard Time on 30 July 2025. MMMMMMMMMMMM MMMMMMMMM 000001 MR A SAMPLE DESIGNATION (IF ANY) ADD 1 ADD 2 ADD 3 ADD 4 ADD 5 ADD 6 ENDORSEMENT_LINE SACKPACK C
1234567890 C O Y MMMMMMM
Here’s how to order a copy of the proxy materials and select delivery
preferences: Current and future delivery requests can be submitted using the options below. If you request an email copy, you will receive an email with a link to the current meeting materials. PLEASE NOTE: You must use the number in the
shaded bar on the reverse side when requesting a copy of the proxy materials. Internet – Go to www.envisionreports.com/ghrs. Click Cast Your Vote or Request Materials. Phone – Call us at 1-866-641-4276 if in the US or Canada or 781-575-2377
if outside the US or Canada. Email – Send an email to investorvote@computershare.com with “Proxy Materials GH Research” in the subject line. Include your full name and address, plus the number located in the shaded bar on the reverse side,
and state that you want a paper copy of the meeting materials. To facilitate timely delivery, all requests for a paper copy of proxy materials must be received by 21 July 2025. Annual General Meeting Notice GH Research PLC’s Annual General
Meeting of Shareholders will be held on July 31, 2025 at Joshua Dawson House, Dawson Street, Dublin 2, D02 RY95, Ireland at 2:00 p.m. Irish Standard Time. Proposals to be voted on at the meeting are listed below along with the Board of
Directors’ recommendations. The Board of Directors recommends a vote FOR all the nominees listed and FOR Proposals 2, 3 and 4: Election of Directors: - Florian Schönharting - Michael Forer, LL.B. - Dermot Hanley - Duncan Moore, PhD To
ratify the appointment of PricewaterhouseCoopers Ireland as independent auditors of the Company for the year ending 31 December 2025 and to authorise the Board to fix the remuneration of the auditors. To renew the Board’s existing authority
under Irish law to allot and issue ordinary shares. To renew the Board’s existing authority under Irish law to allot and issue ordinary shares for cash without first offering those shares to existing shareholders pursuant to the statutory
pre-emption right that would otherwise apply. PLEASE NOTE – YOU CANNOT VOTE BY RETURNING THIS NOTICE. To vote your shares you must go online or request a paper copy of the proxy materials to receive a proxy card. If you wish to attend and
vote at the meeting, please bring this notice with you.