The organisations below must be considered as 'theoretical'.
The Hungarian army lost almost all of its artillery and around 50% of the combined total of all Hungarian military and civilian motor transport in the Stalingrad disaster. The industrial capacity of Hungary could not replace these losses and Allied bombing of factories ensured that artillery and truck production effectively collapsed late in mid-1944.
I can see no evidence that the Germans supplied either the 15cm sFH18 or the 10cm sK18 to Hungary, though the Hungarians had purchased 10.5cm leFH18s and SdKfz 11 tows prior to the war (called the 37M Goring howitzer by the Hungarians). Whether or not any 37Ms survived Stalingrad is not recorded in the sources I have.
The Hungarian choice of heavy howitzer was their own 149mm 14/35.M or 14/39.M. Their light howitzers were 100mm 14/a.M or 14/b.M models.
The majority of these batteries were towed by six-horse teams. The shortage of motor vehicles may have meant that supposedly motorised batteries were also horse drawn too.
The artillery component of a 1944 Infantry Division was;
3 artillery battalions each of;
2 light artillery batteries of 4 horse drawn 10cm howitzers
1 heavy artillery battery of 4 horse drawn 14.9cm howitzers
and
1 motorised heavy artillery battalion of;
2 motorised artillery batteries of 4 14.9cm howitzers
For use as artillery tractors Hungary bought and produced a number of different vehicles;
SdKfz 11 – as mentioned above
Pavesi P4-100 and P4-100 28M
Breda 32
RSO – a few obtained from Germany late in the war
vehicles developed from agricultural tractors such as the Hofherr KV-40 and KV-50.
Quite what still existed, and where, in 1944 is not recorded. In common with most Axis countries I think that whatever was available would be used, and who can say that what you choose to put in your army is wrong?