I guess I was wrong but I don't consider a person wanting to buy a new lawn mower a real need when I am struggling to pay my bills.
I don't know the full situation, however, let me offer a possible scenario to consider. Most likely when you heard the church was buying someone a lawnmower, you got the picture of the average home owner whose lawn mower broke down and they wanted a new one. Maybe that was the situation.
However, maybe the situation is the member earned a living by mowing people's lawns, and his only mower unexpectedly broke down - and that job was their only means of income. Perhaps they also did charity work by mowing church grounds and lawns of elderly people for free on top of their business to pay the mortgage and put food on their family's table. Perhaps there was an unexpected family issue or home repair that strained their finances so when the lawn mower broke, the bank denied them a loan to buy a new one. Thus with no lawn mower, there would be no income, their business may collapse. Also if they did charity lawn mowings, those receiving that benefit would lose out increasing the impact of the loss of the mower.
If the latter was the situation, would this make the donation seem more beneficial?
It's always wise to know what you are donating toward to decide whether your money will go to benefit the cause requesting it. In this situation, the question is whether or not you trust the church's leadership, when they make special donation requests like the above, that they've assessed the situation and have determined it to be a legitimate need based on the individual's situation. If you do, then regardless of what the item the donation is for, trust the money you donated is benefiting the individual in a way you may not understand. If you don't trust their judgement, then revise how/when you donate in future situations like these.