Why is 把 in 我去洗把脸 (wǒ qù xǐ bǎ liǎn) = I'm going to wash my face? YouDao has an example of 我去洗把脸:


  知道了，妈。我去洗把脸，让自己清醒一些。
  Zhīdàole, mā. Wǒ qù xǐ bǎ liǎn, ràng zìjǐ qīngxǐng yīxiē.
  I see, Mummy. I'm going to wash my face and wake myself up.


and I just encountered it in the Chinese TV series 如果蜗牛有爱情, with the English title When a Snail Falls in Love (Wikipedia), in the first episode on QQ about 15 minutes, 49 seconds in:


  　


There's similar examples at YouDao for 洗把脸.

It seems weird to me to put 把 in 洗把脸.  I would expect 我去洗脸 or 我洗脸去.  YouDao indicates 我要去洗脸了 and 把脸洗 are good choices.

Question: Why is 把 in 我去洗把脸?

I see no real reason for 把 to be in there.  My guess is that it's a variation on 把脸洗.
把 is used as 量词 (measure word) here.

三、“把”还可以表示动作的次数和量。大部分用于手的动作，如：拉、帮、推、拽、搭、抓等，后面都可以用“把”，用法跟“次”“下”“回”相近。
可是，在现代汉语中，表示动作的量的意义有所扩展，这些动作与手关系不大。
如：表现一把、玩一把、练一把等，时下最流行的一个词就是“过把瘾”。

It's usually used in spoken language, has similar meaning with 次/下. So 洗把脸 is similar to 洗一次脸/洗一下脸, but sounds more vivid and direct.