depends. does he always say in jocular way "i'll slit my wrists" - do you think he joking or really means it? cry for help? attention seeking? jsut wants a listening ear and chance to moan?
if he is clinically depressed and might "do2 something then only a profressional can help him. but he has to want to get help.
i will tell you what i wish i had done when my then P started saying this stuff:
if he does slit his wrists then get him to A&E and force him to address it. whatever you do, don't cover it up (not easy when they beg you not to tell anyone, but trust me, telling on him is really the best way).
(there are views on self harm eg can be coping mechanism but if he choses to do it so you (or dcs) see it - then he is attention seeking and you need to act...)
if bleeding badly dial 999. if superficial, takes pics adn inform his GP next week...
he is telling you because he wants attention/sympathy etc.
plan some fun times for you and dcs.
give him the option to join in or not. if he chooses to join you, he ahs to make effort to be jolly. then he can have time off eg evening to wallow and moan about life and work etc.... we all need a break to moan/get things off our chest - but you can decide how and in what form so it doesnt take over whole family. eg when kids in bed you sit down with glass of wine and he gets to pour out his problems to you and you listen....
you can support him - but you can decide wwhen and how and in what form that takes.
if you and dcs live according to his depression/anger/angst the next few days/weeks then it wont do him any good and wont do you/dcs any good.
if ehe really cannot snap out of it for dcs sake (separate work and life/family) and your sake for some of the time, then he does seriously need professional help - (counsellor at work?) and you need to make that clear to him. if he does not wish to take help, he has to go elsewhere until he decides to do so.
i have had someone say to me "my wife told me: go get some help or you leave". he knew she would follow thru....he chose to get help. they still together.