Relationship advice from men and women please.?

My partner has been spending many many hours online over the last few months until early hours of the morning.I have brought this up and tried to discuss it like adults and he carried on spending many hours online.He did improve, he started coming to bed between 12 and 2am instead of 3am-4am.However I know i shouldnt have done but I read his messages on the computer, I was hoping to put my mind at ease about him flirting with other women online which has been an issue in the past. I found messages between him and another women that were of a ual nature.One message from this woman said that she was sorry she missed his call.I tackled him about this and he says that he has phoned her about 3 times.I was ready to leave and packed my things to go.I didnt go as we have 2 children aged 3 yrs and a 15 month old and wanted to plan things properly with the least upset to them, I wanted time to sort a house out for me and my two boys.My partner swears that he will not do this again and that it was just abit of fun.He also said that he did it because i have been miserable for two years.I have been very down for the last two years as i lost my mum aged 56yrs two years ago, and my dad aged 54 just one year ago.I aknowledged that i was down and saw a doctor and i am going to take up bereavement councilling.I am confused, should he have supported me for so long over two years or is it understandable that he has done what he has done.He said i never want to sleep with him but this is because i knew about all the online flirting he was doing.One message said i just changed my shirt and shoes so looked like i was fixing the car, i cant help but thing that he has slept with her after that mesage, but he swears that he has never slept with her and she also lives at the other side of the country.I am so confused hurt and angry, so much has happned over the last 2 years i dont know whats acceptable anymore, what to believe.I do know that if i do leave I will be just fine as long as i have my 2 boys and my famiy and friend.Any helpfull advice please.Can things be fixed, should they be fixed or shoul i just enjoy being on my own with my little boys