The story of Mother Sita's banishment by Sri Rama present in the Uttara Khanda of the Valmiki Ramayana, is perhaps the most controversial part of Sri Rama's story. Many Hindu acharyas such as Swami Ramabhadracharya and scholars have argued that the story of Maa Sita's banishment is an interpolation, added centuries later. There are some scriptural arguements that can be made for both sides of course but that's not the subject of my question. I want to know whether any texts other than Valmiki's Ramayana mentions Mother Sita's second exile? I know neither the Ramcharitmanas nor the Mahabharata mentions this story but what about the Puranas? Do they mention Mother Sita's second exile?
Do any puranas mention about Sita's second excile?
Yes , as you have enquired ,Puranas do talk about mother Sita's second excile. Although not in detail as Valmiki Ramayana because it is Itihasa (true history) and Primarily a story of Lord Rama and mother Sita alone. Puranas are mythology (true tails) and talk about various other topics also ,rather than a tail of single god they contain a whole lot of topics related to all the gods and goddesses other narrations and discuss other philosophies of Hinduidm.
For instance take a look at below verse no. 10 of - Skandha 9- Chaptet 11 from Shrimad Bhagvat Purana .Shree Vyasa deva is mentioning a brief account of Shree Rama's life in two chapters 10-11.
He is mentioning in the verse that mother Sita was abandoned (banished) due to fear of people by Shree Ramachandra.
इति लोकाद् बहुमुखाद् दुराराध्यादसंविदः ।
पत्या भीतेन सा त्यक्ता प्राप्ता प्राचेतसाश्रमम् ॥10॥
iti lokād bahu-mukhād durārādhyād asaḿvidaḥ patyā bhītena sā tyaktā prāptā prācetasāśramam
Sukadeva Gosvami said: Men with a poor fund of knowledge and a heinous character speak nonsensically. Fearing such rascals, Lord Ramacandra abandoned His wife, Sitadevi, although she was pregnant. Thus Sitadevi went to the asrama of Valmiki Muni.
अन्तर्वत्न्यागते काले यमौ सा सुषुवे सुतौ ।
कुशो लव इति ख्यातौ तयोश्चक्रे क्रिया मुनिः ॥11॥
antarvatny āgate kāle yamau sā suṣuve sutau
kuśo lava iti khyātau tayoś cakre kriyā muniḥ
When the time came, the pregnant mother Sitadevi gave birth to twin sons, later celebrated as Lava and Kusa. The ritualistic ceremonies for their birth were performed by Valmiki Muni.
Apart from that Padma Purana -Volume 5- PATALAKHANDA - Chapter No. 56 - Rama seeks Bharata's advice. is also mentioning discussion between Shree Rama and Bharata , where Rama orders Bharata to abandon sita in the forest.
From which we came to know about mother sita's second excile.
61-64 O brother , whatever you say is all just. But by my order ,carry out the words which i utter.I know she is purified in fire , is chaste and is honored by the world.(But) I am afraid by public scandal.(Therefor) I (shall) abandon my (wife) Janaki .So taking in your hand sharp and very fiearce sword ,(either) cut off (my) head or abandon (my) wife Janaki in the forest. Hearing these words of Rama ,Bharata with tremor in his body and tears (in his eyes) fell unconcious on the ground.
In the next chapter i.e.57 of Padma Purana Vol.5 we can also find the true reason behind why Rama and Sita were seprated in Sita's second excile
Previous answers presented excerpts from Shrimad Bhagvat Purana, Padma Purana and Tulasidas Ramcharitamanas. So, I am giving an answer using some other scriptures.
Mentions about Rama leaving Sita is present across several scriptures. I am presenting few among them, assuming the intent of question is (just) about mention of Rama leaving Sita due to the rumors on Sita in the kingdom rather than the presence of whole story.
Skanda Purana has multiple mentions on this.
Rama tells it in conversation with vanaras
The monkeys said: O Raghunandana, you appear devoid of all your remarkable splendour! You are very lean and you seem to be distressed. Wh hope,everythingis all right there in your abode. We hope, you are physically all right. Laksmana, your borther, who used to follow you closely is not seen near you now? What has happened to you, O Raghava? Further, your favourite wife, as dear to you as your very vital air, is not seen at your side? We are very much anxious.
Suta said: On hearing their words, Raghava heaved a long and heavy sigh. With tears in his eyes he told them everything: O excellent Brahmanas, he told them why Sita was banished and brother Laksmana abandoned. He told them why he himself had come. On hearing it, all the monkeys with Sugriva as the chief became highly miserable and they cried. They embraced each other for consolation.
Rama tells it in conversation with sage Agastya
On being asked by the sage (Agastya), Rama narrated in detail all the events connected with himself and his departure to heaven. He narrated everything about how Sita was banished by himself, how Laksmana gave up his life on being exiled by himself.
Garga Samhitha also has a story of Rama leaving Sita.
Shri Narada said: O king of Videha, in Treta-yuga, in Lord Rama's kingdom, in Ayodhya, as Lord Rama's spies overheard, this same washerman said to his wife: I will not maintain you, a sinner that returns at dawn from another's house. Ramacandra, who is greedy to enjoy with His wife, may continue to maintain His Sita, but I am not like Him. Hearing from many mouths that the people were criticizing Him, Lord Ramacandra at once exiled Sita to the forest. Lord Ramacandra had no wish to punish him, and in due course this person became a washerman in Mathura at the end of Dvaspara-yuga. To stop his blasphemy, however, Lord Krishna killed him. Then the Lord, who is an ocean of mercy, gave him liberation.
Maa sita's banishment is mentioned in ramcharitamanas as well.
In ramcharitamanas-balkand sag tulsidas says:
बंदउँ अवध पुरी अति पावनि। सरजू सरि कलि कलुष नसावनि॥
प्रनवउँ पुर नर नारि बहोरी। ममता जिन्ह पर प्रभुहि न थोरी॥1॥
सिय निंदक अघ ओघ नसाए। लोक बिसोक बनाइ बसाए॥
बंदउँ कौसल्या दिसि प्राची। कीरति जासु सकल जग माची॥2॥
I reverence the exceedingly holy city of Ayodhyå ( Råmaís birth-place) and the river Sarayu (flowing beside it), which wipes out the sins of the Kali age. Again, I bowto the men and women of the city, who enjoy the affection of the Lord in no small degree. Even though they were damned as a result of the heap of sins incurred by the calumniators of Sitå (who were instrumental in bringing about Her lifelong exile), they were lodged in a heavenly abode, having been divested of sorrow