This is the 17271st most frequent Russian word.
супружеский
"Супружеский" means "pertaining to marriage or spouses."
Here, 'супружескую' is an adjective describing 'жизнь' (life) to denote that it pertains to married life.
Они прожили долгую супружескую жизнь, полную любви и уважения.
They lived a long married life full of love and respect.
In this context, 'супружеские' is an adjective modifying 'отношения' (relationships), referring specifically to those between married partners.
Супружеские отношения требуют взаимного доверия и понимания.
Marital relationships require mutual trust and understanding.
Here, 'супружеские' is used to describe 'трудности' (difficulties), specifying that the challenges are related to their marriage.