English equivalent for the Persian idiom "send someone out in pursuit of black chickpeas"

"To send somebody out in pursuit of some black chickpeas" is a Persian idiom that implies 'to make or ask someone to run an errand so that you be able to have/ buy some time in order to deal with your private affairs in their absence', like having or continuing a (secret) conversation with someone else.

According to its etymology, since 'black garbanzos/chickpeas' were rarely available in the markets, when someone wanted to buy them, they had to search many stores for a long time, and most of time they couldn't find them at all.

Today, its meaning has changed somehow, and it doesn't necessarily mean that we ask someone to run an impossible errand, but it means we use that errand as an excuse to send them away for a while just to have time to do something in their absence.

Example scenario:

Suppose your classmate has come to your house and you are talking about some girly issues in the sitting room. Now you want to tell her a big secret about one of your classmates, but not in the presence of your younger sister who is sitting there too. So you ask her to go to the market and buy some snacks for you. In this way, you will have some time to continue your conversation until she is back. So you might say to your friend:

Well, I sent my sister out in pursuit of some black chickpeas. I'm sure that she won't be back for another 15 minutes, so we can talk more freely now."

And your sister, before leaving the house, might sarcastically say to you:

Okay, but I know this is just an excuse, and you are just sending me in pursuit of some black chickpeas!!! so don't think that you can fool me!

Is there any idiom, expression or proverb that conveys the same meaning or connotation in English (i.e. to make or ask someone to do an errand just in order to buy time so that you can deal with some of your private affairs)?

I have found to be/go on 'a wild-goose chase' but I'm not sure if I can use it or not.

Wild Goose Chase

a search that is completely unsuccessful and a waste of time because the person or thing being searched for does not exist or is somewhere else:

Further reading: The first recorded citation is from Romeo and Juliet, 1592:

Romeo: Switch and spurs, switch and spurs; or I'll cry a match.

Mercutio: Nay, if thy wits run the wild-goose chase, I have done, for thou hast more of the wild-goose in one of thy wits than, I am sure, I have in my whole five.

Our current use of the phrase alludes to an undertaking which will probably prove to be fruitless - and it's hard to imagine anything more doomed to failure than an attempt to catch a wild goose by chasing after it.

Our understanding of the term differs from that in use in Shakespeare's day. The earlier meaning related not to hunting but to horse racing. A 'wild goose chase' was a chase in which horses followed a lead horse at a set distance, mimicking wild geese flying in formation.

Additionally if you are looking to send someone after a specific item, you could try a

Left-handed monkey wrench

a non-existing tool

In a sentence: "Hey Jim, go get me that left-handed monkey wrench".

Wrenches in general do not need a specific orientation or hand to operate and therefore don't need to directional qualifier. Also, other hand tools are designed for right-handed operation as the majority of the population is aligned that way.

See also: pipe stretcher, blinker fluid

The "formal" word in English is pretext:

A reason given in justification of a course of action that is not the real reason

http://www.oxforddictionaries.com/definition/english/pretext

You could say "I sent my sister away on a pretext so we can talk more freely".

Or if you asked your sister to go out and buy something, she might say "I know that's just a pretext to get rid of me for a few minutes."

Although not an idiom, you could maintain the notion of “sending someone on an errand” (without implying that it is an impossible task) by modifying the errand with an adjective that could imply that its real purpose was to cause the errand-runner to “[temporarily] get lost.”
For example:

contrived errand
(used in The Century, Volume 3 via Google Books, where “Mis’ Wilder … banish[es] ‘the girls’ in [a] clumsily contrived errand to the buttery [in order to have some time to gossip with Mis’ Philbrick]”)

Or

concocted errand

“They were down in Gunter's quarters a few minutes later when the caretaker was dismissed on some concocted errand so that the two had the room to themselves.”
(from The Blue Daffodil by Fred M. White, via ‘Project Gutenberg Australia’)

“Well, I sent my sister out on a contrived/concocted errand. I'm sure that she won't be back for another 15 minutes, so we can talk more freely now."

“Okay, but I know this is just a contrived/concocted errand!!! So don't think that you can fool me!”

In New England you can say that you sent a child or children somewhere "to get them out of our hair". The implication is that the children are creating noise, distractions, or making demands which make adult conversation or work difficult.

Wild Goose Chase, above is probably the most correct, but you could also use the related "Snipe Hunt."