In these three (95577, 11774 and 21536) fatwas from islamqa the author or mufti tries to proove that covering the face is the correct view of Islam. However one could find ahadith and Quran rulings that clearly oppose this view or at least don't support it. I'd like to add that instead of starting with a sahih narration or a Quran verse the argumentation there often starts from a statement of scholars and later quotes of more or less sane narrations and interpretations of given Quran verses, which add to the verses and narrations things they don't clearly say!
To me it is strange how or why scholars often start the Issue of hijab/niqab etc. or any kind of covering the body using the verse (24:31)... the verse itself shows that it is only the 2nd part of a message while the earlier part is revealed one verse before (24:30) and addresses men not women! Why? Maybe because men are the stronger part in this "equation" so if they don't stare at what they shouldn't and wouldn't see if they lower their gaze, the chance for anything sinful to happen (at least by force) is much lower!
And if Allah commands men to lower their gaze, what is the use of the order of covering faces of women than?
This is one of the Questions addressed by sheikh Muhammad al-Ghazaly (to get an idea of his view of Islam I strongly recommend you to read at least the chapter about his writings on the Wikipedia link!).
Arguments against covering the face
I'll try to post his argumentation here translated in my own words based on his book "As-Sunna an-Nabawiya bayna ahli-l-Fiqh wa ahli-l-Hadith السنة النبوية بين أهل الفقه وأهل الحديث" (Prophetic tradition between the people of Fiqh (Jurisprudence) and the people of Hadith) (page 55ff, 6th Edition):
قرأت كتيبا في إحدى دول الخليج يقول فيه مؤلفه: إن الإسلام حرم الزنا! وإن كشف الوجه ذريعة إليه، فهو حرام لما ينشأ عنه من عصيان!.
I've read in a booklet in one of the Gulf states where the author says: Islam forbid Zina! And showing the face is a medium towards this and therefore it is haraam because of disobedience which can arise from it.
قلت: إن الإسلام أوجب كشف الوجه في الحج، وألفه في الصلوات كلها، أفكان بهذا الكشف في ركنين من أركانه يثير الغرائز ويمهد للجريمة؟ ما أضل هذا الاستدلال!.
I said: Islam ordered uncovering or showing the face for Hajj and made it familiar for all the prayers. Does it by those two elements awakens instincts and paved the way for the crime? What a misguided argumentation is this?
وقد رأى النبي ـ صلى الله عليه وسلم ـ الوجوه سافرة في المواسم والمساجد والأسواق فما روي عنه قط أنه أمر بتغطيتها، فهل أنتم أغير على الدين والشرف من الله ورسوله؟.
And the Prophet (صلى الله عليه وسلم) has seen the face unveiled in many occasions, in the mosque, in the market. And no order of covering has been narrated on his authority. Do you fear for the religion and honor more than Allah and His Prophet?
And he quoted a couple of verses and hadiths with commentaries to give a proof:
After quoting verse (24:31) he asked should the believers lower their gazes to not take a look at the poll and the neck?
الغض يكون عند مطالعة الوجوه بداهة، وربما رأى الرجل ما يستحسنه من المرأة فعليه ألا يعاود النظر عندئذ كما جاء في الحديث.
Lowering the gaze is by intuition when looking at the face, and maybe a man may see what he likes in a woman's face, so he shouldn't do it again as stated in the hadith.
And what to do if a men's instincts by an accidentally look awaken is quoted in this hadith, in this case the married men should be happy with what he has (his wife). While non-Married men should follow (24:33):
But let them who find not [the means for] marriage abstain [from sexual relations] until Allah enriches them from His bounty. ...
حكى القاضي عياض عن علماء عصره ـ كما روى الشوكاني ـ أن المرأة لا يلزمها ستر وجهها وهي تسير في الطريق، وعلى الرجال غض البصر كما أمرهم الله...
Al-Qadi 'Iyad has narrated from the scholars of his epoch -as a-Shwakani has narrated- that a woman doesn't need to hide or cover her face while walking in the streets while men are asked to lower their gaze as Allah has ordered them!
On a 'Id our Prophet () hold a sermon addressing women (note that the Mosalla -the place where the 'id prayer is held- is frequented by both genders as ordered by our Prophet) and preached to them and admonished them, and asked them to give alms, for most of them are the fuel for Hell (see for example this hadith). In this hadith the narrator said: "A woman having a dark spot on the cheek stood up", sheikh al-Ghazaly here asked how would he know the color of her cheek if she was face-covered? An other narration of the same hadith also refutes the idea of covering the hand indirectly! While al-Ghazaly say that there's a version saying that Jabir even saw them taking of rings etc. which would be even clearer on the matter!
قال بعض الناس: إن الأمر يكشف الوجه في الحج، أو في الصلاة، يعطى أن الوجه يجب ستره فيما رواء ذلك، وأن على المرأة ارتداء النقاب. والقفازين!.
Some said: That the order to show the face during Hajj or during the prayer implies that they should be covered in any other situation. And women should wear niqab and gloves
ونقول: هل إذا أمر الله الحجاج بتعرية رؤوسهم في الإحرام كان ذلك يفيد أن الرءوس تغطي وجوبا في غير الإحرام؟ من قال ذلك؟ من شاء غطى رأسه ومن شاء كشفه..
We say: If Allah ordered to uncover heads during Ehraam in Hajj would that mean that we are to cover our heads outside of Ehraam? Who said so? One is free to cover or uncover his head.
He also quoted this hadith (see also these versions 1,2,3,4). Then he asked:
والسؤال فيم صعد النظر وصوبه إن كانت منقبة؟.
The Question here is at what he () looked at if she was wearing a niqab before lowering his head?
Result or conclusion and advice
I could also quote other examples (in total 11 examples and refuted counter-examples were quoted, I only posted 5 of them) from the book, but I'd like to come to a conclusion. Sheikh al-Ghazaly later showed that the view on which most scholars are -more or less- in "consensus" about is that the face should be uncovered and the hands too based on this hadith. And this by clear statements of authorities of the four sunni madhabs and most scholars like at-Tabari, Sa'id ibn Jubayr, ad-Dahhak (not sure about the linked Biography), al-Awza'i and the salafi -attributed- scholar ibn Kathir (who said it is the opinion of the majority of the scholars) however the hanbali school say that they have two opinions/views about the hands, while the hanafi even allows uncovering the feet.
As you see covering your face doesn't have strong backup in Allah's shari'a.
And therefore if you feel uneasy or disturbed by people who plop in the room you are praying and see you pray, which may mean that they see you performing some "strange" bows or exercises. You should either let them know about it, seek another room or pray in a "hidden" corner. I'm afraid that covering your face won't be a good idea as I assume you are in a non-Muslim society!