The commandment, tenth in that list Moses received on the mountain (Exodus 20:17) is:
You shall not covet your neighbour's house; you shall not covet your neighbour's wife, or male or female slave, or ox, or donkey, or anything that belongs to your neighbour. (NRSV)
In other words, you shall not covet that which does not belong to you. So there's no danger of a woman coveting her neighbour's husband, because men don't 'belong' to women ... sorry, that's another blog post.
The Hebrew translated as 'covet' here, means to desire and try to obtain. The English word 'covet' carries the meaning of yearning to possess. The prohibition really is about property. Respect the property of your neighbour, avoid jealousy, greed, theft, deception, which are all things that flow from desiring and trying to obtain that which does not belong to you.
When someone 'covets' my prayers, then, perhaps they are desiring and hoping to obtain my prayers. But my prayers are not my 'property'; my prayers are available for offering on your behalf. Even so, can you really 'possess' my prayers? It would seem, then, that there is a semantic inadequacy about this language choice.
We use language to communicate. To say what we mean, however, is only half the equation in communication. To say something that will be received with the meaning we intend – now that is the aim. There is no denying that when someone uses 'covet' language, it carries more baggage than a desire and hoping to obtain prayers that are perfectly ok to desire and hope to obtain (though not 'possess'). So long has 'covet' been used in a negative sense, yearning to possess something that belongs to someone else, that to use it to express a request for the positive gaining of prayerful support and solidarity seems incongruent with its received meaning over time. It would also seem, then, that there is an overlooking of the cultural meaning received on hearing 'covet', when this word is chosen.
I would like to encourage those who request prayers in the manner of 'coveting' prayer, to choose different language. I do not think you are saying what you mean; or at least, meaning what you are heard to say.