Really you should be using a vector with an offset. Or even an array with an offset. The extra addition or subtraction isn't going to make any difference to the speed of execution of the program.
If you want something with the exact same speed as a default C array, you can apply the offset to the array pointer:
int* a = new int[10];
a = a + 5;
a[-1] = 1;
However, it is not recommended. If you really want to do that you should create a wrapper class with inline functions that hides the horrible code. You maintain the speed of the C code but end up with the ability to add more error checking.
As mentioned in the comments, after altering the array pointer, you cannot then delete using that pointer. You must reset it to the actual start of the array. The alternative is you always keep the pointer to the start but work with another modified pointer.
//resetting the array by adding the offset (of -5)
delete [] (a - 5);