A loan is most generally a liability, a part of the balance sheet. Expenses & income are part of the income statement. Income is the net of revenues after expenses.
The interest is an expense on the income statement, but the loan itself does not reside there unless if it is defaulted and forgiven. Then it would become a revenue or contra-expense, depending on the methodology.
The original purpose of the income statement is to show the net inflows of short term operational accruals which would exclude new borrowing and repaid loans.
The cash flow statement will better show each cash event such as borrowing debt, repaying debt, or paying off a bill.
To show how a loan may have funded a bill, which in theory it directly did not because an entity, be it a person or business, is like a single tank of water with multiple pipes filling and multiple pipes extracting, so it is impossible to know which exact inflow funded which exact outflow unless if there is only one inflow per period and one outflow per the same period.
That being said, with a cash flow statement, the new loan will show a cash inflow when booked under the financing portion, and paying a bill will show a cash outflow when booked under the operating portion. With only those two transactions booked and an empty balance sheet beforehand, it could be determined that a new loan funded a bill payment.