Does correspondence to any of the Ottoman caliphs remain?

A number of pieces  of correspondence remain. I cite a couple of examples in this answer.  Letters written to the Emperor especially during the reign of Humayun to 'Suleiman the Magnificent' illustrates this fact: 'Gifts of sincere wishes to your exalted majesty the possessor of the dignity of Khilafat, the pole of the sky of greatness and fortune, the consolidator of the foundations of Islam. Your name is engraved on the seal of greatness and in your time the Khilafat has been carried to perfection. May your Khilafat be perpetuated. God be praised that the gates of victory are opened by the eyes of His inspiration and His dispensation the seat of the Sultanate and the throne of the Khilafat of the realms of Hind and Sind is once again graced by a monarch (Humayun) whose magnificence is equal to that of Sultan Suleiman the magnificent.' (Naimur Rahman Farooqi, A Study of Political and Diplomatic Relations between Mughal India and the Ottoman Empire,1556-1748, Volume 1) This is a letter sent by Hajj Mohammad Doul. He founded London's first masjid asking the Ottoman Caliph for funds for the work he was doing. The letter transcribed reads: To Your Excellency Hadjie Ali Bey May it please your Excellency! May I address the following lines to Your Excellency! On the 17th of January this year I sent Your Excellency a document containing the works I have done and am still doing for the great and noble cause of Islam and for the prestige and honour of our most beloved Calipha, upon whom all good blessings flow! I do herewith enclosed [sic] another document which conveys the same sense as the previous, and which together with the first document would never have been sent to the Seat of the Caliphate had not the expenses I incurred been very, very great during the last five years in London while agitating for the cause of the Calipha and the Ottoman Empire and through which agitation (this being and is still an absolute necessity) my ordinary occupation has been entirely neglected. I, therefore, pray through the Grace of Almighty Allah, your Excellency will be so kind and generous as to lay this said document at the Foot of the Throne of His Most Gracious Majesty our beloved Calipha Al-Kazi Sultan Abdul Hamid Khan, the Victorious. Would it be out of place were I to repeat that when in Constantinople last year (1897) I have had the pleasure and fortune to have been introduced to Your Excellency by His Excellency Marshall Reouf Pasha and that at the interview which took place I was kindly told to submit my document. In the Salaat Travie [Tarawih] which I and the Congregation of the London Temporary Mosque are, of course, performing every evening during the Ramadan a Doa [du'a] is also made for your Excellency's good health, long life and prosperity. May, through the grace of Allah, your Excellency continue rendering the good services for Islam which the Moslem World has always been benefited by! May your Excellency enjoy a happy and successful term of office! May I, on behalf of the congregation of the Mosque, convey our best wishes to your Excellency for the coming Eid [al-]Fitr (Bairam). I have the honour to be Your Excellency's Most Humble and Obedient Servant Hadjie Muhammad Doulie London Temporary Mosque, 97, Albert Street, Regent's Park London N.W. England Source: Ottoman State Archives Y. PRK. TKM. 0040.0023.001 Courtesy: Dr Halim Gencoglu, University of Cape Town