A person who is "broke" has usually been brought to that situation by calamity or sin, or perhaps oppression. A response is relatively simple. But a person who is "poor" has usually been brought to that condition by an interlocking grid of oppression, calamity, and sin. Helping appropriately takes more time, more wisdom, and more effort.
This and lots of other good points are made by Tim Keller in Ministries of Mercy. Grace, says Keller, is not deserved, but it is conditional. Grace demands change; it holds us accountable. How do we do that with fellow humans -- hold this balance between undeserved but conditional? "Grace is undeserved caring that intercepts destructive behavior". (p. 227) Justification moves into sanctification. Our ministry must help people freely, yet aim to bring their whole lives under the healing Lordship of Christ. (p. 227)
Keller suggests that long term help that deals with long-standing patterns must always develop into a willingness to allow access into all of life. The deacon says, in effect, "If we are going to continue to help, you must be willing to let us into your life." What kind of relationship-building leads most likely to this kind help?
A second intriguing idea is "Let mercy limit mercy". When do you stop helping? Keller says there is only one legitimate rationale for stopping: when continued help would shield the person from the consequences of his or her own irresponsible behavior. This takes great discernment, and obviously there are many factors to consider, including HOW the message is given.
This book is not so new - its second edition came out in '97. But it's Keller, and he's worth studying. Its 14 chapters would make a high quality study series for a council, for deacons, or for any group of interested members of your church.