I dont think the fault will be the gas pipe being too short as that wont stop the gas comming through or it shouldnt any way. As if its just a hob then i assume its on a copper pipe and not a flexi hose as you shouldnt use a gas hob on a flexi hose anyway.
As this is a second hand hob if there isnt much gas comming though theres a good chance it has the wrong jets in there. The person that had it last may have had it set to lpg gas which have different jets so if you have it to normal gas then the jets are to small and is restricting the gas comming through.
So from the sounds of this it has the wrong gas jets in there and will need to be swapped over with new jets im afraid.
The jets are roughly about £25 for the set so its probberly cheaper to replace the jets then get a new hob installed.