I have had a tough time beating HRE as Delhi. They go to Castle too fast or they have a ton of MAA or Knights early. So far i have only had luck if i am able to raid their gold early. Other then that nothing. I have tried to fast castle myself and steal all the relics but they are considerably faster.
I would think sacred cites would be your best bet. Get them fast, and build defenses. Spend some extra time in fuedal and punish the castle rush. They shouldnt have many units ready to contest, and by the time they do get some in castle you should be pretty well ready to defend while transitioning to castle yourself, all while getting free time on that victory timer.
Maybe even think about walling them in to buy even more time, if the map allows
With the last patch the things are changed, but some general strat can still works.
I don’t know your Elo but you can try 2 strats against HRE. Stay in feudal and rush him with everything, put scolars inside stable and archery range and raid him fast as you can. Try to deny gold and wood in first. Food is under tc so it’s harder to deny.
Or you can try to take control of the map. With some spears you can go around to walling in relics and take and fortify atleast two holy sites. Behind this you go in castle too
I suggest to familiarize with aoe4world.com , There is a lot of usefull information and lastly on tools you can watch replays from twitter selecting type of civilization against, elo, map…
For example here https://aoe4world.com/tools/twitch-video-finder?civilization=delhi_sultanate&other_civilization=holy_roman_empire&map=&rating=>1100&game_length=
There are some replay Delhi vs Hre from player >1100 elo. In green the winner of the match.
Deny gold and protect relics.
If you deny gold to HRE is over, put some towers in their gold and get sacred sites in feudal.
Obviously if you let him pass to castle and get the relics they will start to spam MAA and knights. Also if they go to palace they will spam 5 MAA in the time they will build 1, so of you are not in castle and ready to build crossbowmen you will get flooded.