The narration you've mentioned shows that among the scholars of Medina covering the mouth was considered as a bad act which shouldn't be done while praying. As Salim ibn 'Abdullah ibn 'Umar is one of the so called 7 scholars of Medina from among the tabi'yn and a grand son of 'Umar ibn al-Khattab.
Even if there are further evidences scholars differed about the reason for this prohibition or at least disapproval:
Al-Kandahlawi in his commentary called Awjaz al-Masalik ila Muwatta' Malik أوجز المسالك إلى موطأ مالك (Volume I pages 337-338 here) said that the Magi's used to pray covering their mouth when worshiping their fire as stated by az-Zaila'y.
While some scholars said it is due to the fact that this way sujud is not complete as the nose and/or forehead wouldn't -directly- touch the ground if the face was covered. But it is mostly explained as matter of modesty in opposition to arrogance.
Generally covering the mouth during the prayer is markruh for both male and female (nevertheless the prayer would be regarded as valid if performed this way). Note that the same applies to tawaaf for women due to the known hadith.
And it is even haram if the intention to cover is due to arrogance.
This actually doesn't mean that one shouldn't do it if there's a necessity: For example an allergy.
This is also supported by other statements and narrations such as:
The Messenger of Allah (ﷺ) forbade trailing garments during prayer and that a man should cover his mouth.
Abu Dawud said: This tradition has also been narrated by 'Isi on the authority of 'Ata' from Abu Hurairah: The Prophet (ﷺ) forbade trailing garments during prayer.
(Sunan abi Dawod)
“The Messenger of Allah (ﷺ) forbade a man to cover his mouth during the prayer.” (Sunan Ibn Majah)
However covering parts of the face (talathum: using al-Litham اللثام or tanaqub: using an-Niqaab النقاب) has different definitions among scholars:
The shafi'i's call talathum covering the mouth. The hanafi's and hanbali's say say it is covering both nose and mouth. And the maliki's say it is to the end of the lower lip.
Ibn Rushd in his al-Bayan wa at-Tahseel البيان والتحصيل even quoted that covering the beard (for men) is makruh (See here page 250) based on the given narration.
Here some views from fiqh books:
In the following I'm translating from Arabic language, as these translations are of my own take them carefully!.
- Among the things which are makruh s mentioned by al-Khatib a-Shirbini (shafi'i scholar) نهاية المحتاج إلى شرح المنهاج :
وأن يصلي الرجل متلثما والمرأة متنقبة
And if a man prayed covering his mouth (mutalathim: using Litham) or a in case of a woman (mutanaqibah: using Niqab)
- And n a-Taj wa al-Iklil التاج والإكليل
يكرهان وتسدل على وجهها إن خشيت رؤية رجل
Both are frowned upon (referring to suing Litham or Niqab for a -free- woman), but she might trail the garment if she fears being seen by a man.
See also islamweb fatwa #52652 in Arabic.