Masayoshi (正義)
The Japanese Wikipedia counts Hosokawa Masayoshi among the known students of Suishinshi Masahide (水心子正秀), with Taikei Naotane, and Tomita Kiyama’s list of shintō swordsmiths (1933) places him in Edo. He signs 作陽幕下士細川正義 (Sakuyō bakka-shi Hosokawa Masayoshi), often with a stamped seal (刻印, kokuin), and once 作陽士細川正義 (Sakuyō-shi Hosokawa Masayoshi). His 1842 katana bears the dedication 依指田孝勝之好鍛之 (Sashida Takakatsu no konomi ni yori kore o kitau), “forged to the taste of Sashida Takakatsu”. Neither his dates of birth and death nor his genealogy are recorded in the sources consulted.
His students
According to the Japanese Wikipedia, Fujieda Hideyoshi (藤枝英義) learned first with his father, Hideichi, and then with Hosokawa Masayoshi; when he received the complete teaching (皆伝, kaiden) from him, he took the name 英義 (Hideyoshi), with the 英 (hide) of his father and the 義 (yoshi) of his master, and in 1853 became swordsmith of the Kawagoe domain. According to Toryu-Mon, Katsumura Norikatsu, of Mito, trained in Edo with Masayoshi and with Unju Korekazu, signed some blade Hosokawa Masanori (細川正徳) and was head of his workshop (塚頭, tsukagashira).
Masamori (正守)
Masamori signs like Masayoshi, 作陽幕下士細川正守 (Sakuyō bakka-shi Hosokawa Masamori), also with a seal, and on a tachi 作陽幕下士細川主馬佐源正守 (Sakuyō bakka-shi Hosokawa Shume no suke Minamoto no Masamori). A katana designated in Tochigi bears both signatures: 作陽幕下士細川正義 同正守造之 (Sakuyō bakka-shi Hosokawa Masayoshi, dō Masamori kore o tsukuru). The joint work and the same title suggest a son or a student of the house, but the sources consulted do not say which.
The Shimotsuke branch
In Shimotsuke other Hosokawa sign. Masahira (正平): 野州住細川正平 (Yashū-jū Hosokawa Masahira), on a katana of August 1840 (Tenpō 11). Yoshinori (義規): 野州住細川義規作 (Yashū-jū Hosokawa Yoshinori saku), and also 宇都宮藩士細川義規 (Utsunomiya hanshi Hosokawa Yoshinori), “retainer of the Utsunomiya domain”. Masanori (正規): 細川義規男正規造之 (Hosokawa Yoshinori no otoko Masanori kore o tsukuru), “made by Masanori, son of Yoshinori”, with one of his katana at the Furumine shrine, in Kanuma. And Morihide (守秀), with a wakizashi signed 細川守秀造 (Hosokawa Morihide tsukuru). How Masahira, Yoshinori and Morihide are related to Masayoshi is not recorded in the sources consulted.