When one recites a Beracha over a cup of wine – such as Kiddush, or when Birkat Ha’mazon is recited over a cup of wine – he should not use a cup that is "Pagum" ("defective"), meaning, a cup from which he or somebody else had drunk. Since some residual saliva is in the cup, it is disrespectful to use that cup as "Kos Shel Beracha" – a cup over which a Beracha is recited. Hacham Bension Abba Shaul (1924-1998) ruled (in Or Le’sion 20:11) that this applies even to a cup from which one drank using a straw; even though his mouth never touched the cup, the cup is nevertheless considered "Pagum."
One can "fix" a cup that is "Pagum" and make it suitable as a "Kos Shel Beracha" by adding some water or wine into it. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) noted (in Parashat Shelah) that this is effective in making the cup suitable only if the cup from which one pours into the "Kos Shel Beracha" is itself not "Pagum."
If one is reciting Kiddush, and immediately after reciting the Beracha of "Gefen" he mistakenly sipped some of the wine, before the reciting the Beracha of Kiddush, he now faces a Halachic dilemma, as the cup is considered "Pagum." Hacham Bension writes that if the person can quickly add some water or wine to the cup, he should do so. But if this would cause a delay in the middle of Kiddush, he should simply continue, even though the cup is "Pagum," because the Shulhan Aruch writes that after the fact, a Beracha recited over a cup that is "Pagum" is valid. This is preferable to making a delay in the middle of Kiddush.
If a "Kos Shel Beracha" is not filled to the top, but it contains the minimum required amount of wine (a Rebi’it), one may place a small piece of bread into the cup so that the wine will rise to the top.
Summary: When one recites Kiddush or Birkat Ha’mazon over a cup of wine, he should not use a cup from which he or somebody else had drunk, either directly or with a straw. If somebody had drunk from the cup, it can be "fixed" by adding some water or wine to the cup. If one mistakenly drank from the cup during Kiddush, he should add some wine or water to the cup, if this can be done quickly. Otherwise, he should simply proceed with Kiddush.