Problem with Caml Query

You need to add an extra <And>/<Or> clause in the <Where>.
All lines with 'New' in front of them are the ones you need to add.

     for (var i = 0; i < NewsLists.length; i++) {
            var soapEnv = "<soapenv:Envelope xmlns:soapenv='http://schemas.xmlsoap.org/soap/envelope/'>";
            soapEnv += "    <soapenv:Body>";
            soapEnv += "        <GetListItems xmlns='http://schemas.microsoft.com/sharepoint/soap/'>";
            soapEnv += "            <listName>" + NewsLists[i].ListGUID + "</listName>";
            soapEnv += "            <query>";
            soapEnv += "                <Query>";
            soapEnv += "                    <OrderBy Override='TRUE'>";
            soapEnv += "                        <FieldRef Name='Created' Ascending='FALSE' />";
            soapEnv += "                    </OrderBy>";
            var currentDate = new Date();
            currentDate = currentDate.toISOString();
            soapEnv += "                    <Where>";
New         soapEnv += "                        <And>";
            soapEnv += "                            <And>";
            soapEnv += "                                <Or>";
            soapEnv += "                                    <IsNull>";
            soapEnv += "                                        <FieldRef Name='DateTo' />";
            soapEnv += "                                    </IsNull>";
            soapEnv += "                                    <Leq>";
            soapEnv += "                                        <FieldRef Name='DateFrom' />";
            soapEnv += "                                        <Value Type='DateTime' IncludeTimeValue='True'>" + currentDate + "</Value>";
            soapEnv += "                                    </Leq>";
            soapEnv += "                                </Or>";
            soapEnv += "                                <Or>";
            soapEnv += "                                    <IsNull>";
            soapEnv += "                                        <FieldRef Name='DateFrom' />";
            soapEnv += "                                    </IsNull>";
            soapEnv += "                                    <Geq>";
            soapEnv += "                                        <FieldRef Name='DateTime' />";
            soapEnv += "                                        <Value Type='DateTime' IncludeTimeValue='True'>" + currentDate + "</Value>";
            soapEnv += "                                    </Geq>";
            soapEnv += "                                </Or>";
            soapEnv += "                            </And>";
New         soapEnv += "                            <Eq>";
New         soapEnv += "                                <FieldRef Name='PublicY' />";
New         soapEnv += "                                <Value Type='Choice'>Yes</Value>";
New         soapEnv += "                            </Eq>";
New         soapEnv += "                        </And>";
            soapEnv += "                    </Where>";
            soapEnv += "                </Query>";
            soapEnv += "            </query>";
            soapEnv += "            <queryOptions>";
            soapEnv += "                <QueryOptions>";
            soapEnv += "                    <IncludeMandatoryColumns>TRUE</IncludeMandatoryColumns>";
            soapEnv += "                </QueryOptions>";
            soapEnv += "            </queryOptions>";
            soapEnv += "            <viewFields>";
            soapEnv += "                <ViewFields>";
            soapEnv += "                    <FieldRef Name='ows_Title'/>";
            soapEnv += "                    <FieldRef Name='Ingress'/>";
            soapEnv += "                    <FieldRef Name='Visning'/>";
            soapEnv += "                    <FieldRef Name='Bild'/>";
            soapEnv += "                    <FieldRef Name='ows_FileLeafRef'/>";
            soapEnv += "                </ViewFields>";
            soapEnv += "            </viewFields>";
            var rowLimit = MaxAllNews;
            soapEnv += "            <rowLimit>" + rowLimit + "</rowLimit>";
            soapEnv += "        </GetListItems>";
            soapEnv += "    </soapenv:Body>";
            soapEnv += "</soapenv:Envelope>";