This passage occurs as a RCL "Pentecost"/"Ordinary"/"Proper" Season, Year A, most recently June 2017.
Summary: We are familiar with the Great Commission of Matthew: Go therefore... This week we hear the Least Commission: We are sent to do small things to the least of these.
αποστειλαντα ("send" aorist participle of αποστελλω 10:40) Perhaps you might be familiar with the phrase or concept: "The sending of the Trinity." This idea develops out of verses like this one: The Father sends the Son; who sends the Spirit; who, along with Jesus, sends the disciples. This motif is most recognizable in John (John 3:16 for Father sending son; John 14:26 and 15:26 for the sending of the Spirit; John 20:21 for Jesus sending the disciples.). Luke has a similiar phrase in 10:16: "Whoever listens to you listens to me, and whoever rejects you rejects me, and whoever rejects me rejects the one who sent me" (This verse uses the same participle as Matthew 10:40; Mark 9:37 also is similar). The point of these various Scripture citations is to show that Trinitiarian thinking is deeper in Matthew (and the other Gospels) than we often give credit. Furthermore, one of the earliest ideas of the Trinity was this procession of sending. It is also worth noting that even in Matthew's Gospel Jesus equates action with himself to action to God.
This ties also back to Matthew 10:5 and the sending of the disciples by Jesus.
δεχομενος ("welcome" present participle of δεχομαι; 10:40). This word can also mean accept (See Matt 18:5). Instead of accepting Jesus, you need to accept your pastor, who stands in line of the apostles :-)
μισθος ("wages" or "reward" 10:41,42) I am not sure what to make of it, but Matthew uses this word way more than the other Gospels. I think it might reflect the fact that Matthew aims at the working class, who would be well aware of the reality of wages and rewards? For the most part Matthew is telling others that they already have their reward or that they will not get theirs! In this case though, Matthew has Jesus offer us a promise: If you welcome a prophet, you get your reward; if you give a small cup of water, you also gain your reward. The question remains, what is the reward?
προφητης ("prophet" 10:41) Worth noting: For Matthew, the notion of prophecy is very important; the word appears 34 times. By comparison, in Mark the word only appears 5 times! Luke 28; John 14. It is always worth remembering that connections to the OT are important for Matthew, but Luke doesn't leave them out!
μικρων ("least of these" from μικρος 10:42). This phrase is often understood to mean "children." This is because in Matthew 18 Jesus explicitly connects the phrase little ones with the word for children. Also, Jesus says, "Who welcomes children, welcomes me" in all three synoptics. So, it is probably a fair translation to say, "children" here. However, I think that Matthew 25 and, "Do unto the least of these" is probably a fair direction for understanding this passage too. Jesus is always concerned about the least in society, of which children are an example. I'd rather leave the translation as the "least of these" instead of "children" to leave open this ambiguity. As a side note, some manuscripts use the word "least" that is found in Matthew 25 (ελαχιστος).
Grammar Review: ου μη
In Greek the strongest denial of a possibility is ου μη. It probably best means "It ain't never ever gonna happen." Whenever you see this, you can know the speaker is completely and totally sure about something. In this case, we will never lose our reward when we give a cup of cold water to the least of these.