Thank you for confirming. Firstly, you should be aware that despite the inheritance contribution and your on-going contributions to the property he has a legal interest by being named on the title as well as matrimonial home rights to occupy the property.
However, given that you are the sole carer of both children and you have a low income it would be unreasonable for him to attempt to sell the property to obtain any equity as by the sounds of it you will not be able to afford to move out and a court will not agree to put you and the children in such a position.
In the current circumstances you have good grounds to pursue what is called a Mesher Order, which is an order that you and the children stay in the property until the youngest child reaches a certain age (eg. finishes education), and at that point the property to be sold and equity to be divided.
Unfortunately, despite the apparent lack of financial contributions from him to the property, he would still be able to make a claim towards it due to the marriage and the court's starting position is a 50-50 split of all assets. However, as your needs are greater than his it is likely that any split will be more in your favour.
For your information, when either making an order or approving a settlement the court takes into account the following factors:
The criteria considered is:
1. The income, earning capacity, property and other financial resources which each of the parties to the marriage has or is likely to have in the foreseeable future, including in the case of earning capacity any increase in that capacity which it would in the opinion of the court be reasonable to expect a party to the marriage to take steps to acquire;
2. The financial needs, obligations and responsibilities which each of the parties to the marriage has or is likely to have in the foreseeable future;
3. The standard of living enjoyed by the family before the breakdown of the marriage;
4. The age of each party to the marriage and the duration of the marriage;
5. Any physical or mental disability of either of the parties to the marriage;
6. The contributions which each of the parties has made or is likely in the foreseeable future to make to the welfare of the family, including any contribution by looking after the home or caring for the family;
7. The conduct of each of the parties, if that conduct is such that it would in the opinion of the court be inequitable to disregard it;
8. In the case of proceedings for divorce or nullity of marriage, the value to each of the parties to the marriage of any benefit which, by reason of the dissolution or annulment of the marriage, that party will lose the chance of acquiring.
If you have any further questions regarding this please let me know. In the meantime if you found this information helpful please provide a positive rating using the stars at the top of this page. I will not be credited for answering your question without a positive rating. Thank you