Are relations between a Jewish married man with a non-Jewish woman either married or single considered adultery?
Adultery, or Arayot in Hebrew, in Jewish law is (according to yeshiva.co here)
the prohibited sexual relations a man or woman, married or not, can't have. The list appears in Torah, Vayikra 18 and 20. All those listed, will be punished with "Karet" and in addition, some carry the death penalty by Beth Din as well.
The prohibition is not only on the intercourse but any physical contact of affection and even non-physical affection contact is a rabbinic prohibition.
Among the list for men, are a married woman and a Niddah.
However non-Jewish women are not part of the definition of adultery. The Rambam (MT Issurei Biah 12:2) discusses the punishment for a Jewish man having relations with non-Jewish women
When, by contrast, one engages in relations with a gentile woman with a licentious intent, he is given "stripes for rebellious conduct" according to Rabbinic Law. [This is a] decree, lest this lead to marriage.
In other words relations with non-Jewish women are strictly forbidden for fear they will lead to marriage since intermarriage is forbidden (Devarim 7:3).
Regarding adultery with a Jewish woman, see here. And see also this related question: From where do we know that sexual intercourse with a non-jewish woman is forbidden?
Because Judaism allowed a man to have more than one wife, until a thousand years ago, technically a married man could go find another woman without breaking halacha, and thus the strictest halachic definition of "adultery" only concerns relations between a married woman and a man who's not her husband.
Still, it should go without saying that for a married man to cheat on his wife is despicable, wrong, a violation to fellow human beings, and a sin to God. (At the bare minimum, all non-marital relations are prohibited.) The technical term "adultery" is not used, however.
A series of post-Biblical prohibitions (likely dating back to the Maccabees) exists concerning a Jewish man having promiscuous relations with a non-Jewish woman; see Avodah Zarah 36b. If the non-Jewish woman was married, there's an additional Biblical prohibition: it appears from the commentaries that the traditional thou-shalt-not was intended for relations between Jews as that was far more of a concern than some far-flung people who didn't speak your language; however it violates the thou-shalt of Genesis 2:24, therefore a man shall ... be united with his wife. (See Tos. Kiddushin 21a and Beis Shmuel E"H 16:1)
Short answer -- it's terrible and wrong. Can and does lead to the desecration of God's Name. All legalese aside, Rabbi Emanuel Feldman commented on one high-profile philanderer that a midrash reads: The pagan says "there is none above me." The thief says "there is no one parallel to me." The adulterer says "there is none at all but me."
Some sources to support @Mbloch's answer to understand the Sugya in full (Gemmorah Avodah Zara 36a).
In short, it is a [very severe] Rabbinical decree, since Rabbis compared all gentile women to ritually unclean (נידה) it can be [speculated] as adultery Derabanan. All other comparisons (mistress, gentile or prostitute) do not constitute adultery:
ישראל הבא על העובדת כוכבים הלכה למשה מסיני היא דאמר מר הבועל ארמית קנאין פוגעין בו
The Gemara rejects this: The prohibition concerning a Jew who engaged in intercourse with a gentile woman is a halakha transmitted to Moses from Sinai, not a rabbinic ordinance. As the Master said: With regard to one who engages in intercourse with an Aramean woman, zealots may attack him, as Pinehas did to Zimri in the wilderness (see Numbers 25:6–8).
א"ל דאורייתא בפרהסיא וכמעשה שהיה ואתו אינהו גזור אפילו בצינעא בצינעא נמי בית דינו של חשמונאי גזרו
He said to him: By Torah law intercourse with a gentile is prohibited in public, and only in situations like the incident that occurred, as described in Numbers, chapter 25. And the students of Shammai and Hillel came and decreed that the prohibition applies even in private. The Gemara raises another difficulty: This was also prohibited in private, as the court of the Hasmoneans decreed that it is prohibited.
[דכי אתא רב דימי אמר ב"ד של חשמונאי גזרו] ישראל הבא על העובדת כוכבים חייב משום נשג"א
As when Rav Dimi came from Eretz Yisrael to Babylonia, he said: The court of the Hasmoneans decreed that a Jew who engaged in intercourse with a gentile woman bears liability for transgressing four prohibitions, represented by the mnemonic: Nun, shin, gimmel, alef. These letters stands for: Menstruating woman [nidda], maidservant [shifḥa], gentile [goya], and married woman [eshet ish]. By rabbinic law, a man who engages in intercourse with a gentile woman is considered to have violated the prohibitions involved in having intercourse with all four of these women.
כי אתא רבין אמר משום נשג"ז
And when Ravin came from Eretz Yisrael to Babylonia, he said: He bears liability for four prohibitions represented by the mnemonic: Nun, shin, gimmel, zayin, which stands for: Menstruating woman [nidda], maidservant [shifḥa], gentile [goya], and prostitute [zona]. In any case, it is apparent that this decree was in force before the time of the students of Shammai and Hillel.
כי גזרו בית דינו של חשמונאי ביאה אבל ייחוד לא ואתו אינהו גזור אפי' ייחוד ייחוד נמי בית דינו של דוד גזרו
The Gemara answers: When the court of the Hasmoneans decreed, they prohibited only sexual intercourse, but with regard to seclusion with a gentile woman, no, they did not prohibit that. And the students of Shammai and Hillel came and decreed that even seclusion with a gentile woman is prohibited. The Gemara raises an objection: Seclusion was also prohibited earlier, as the court of King David decreed that with regard to this matter.